Rav Elyashiv - A Talmid Chochom Cannot Be a Baal Gaava
It is told that someone once asked Rav Elyashiv how can there be Talmid Chochom who is a Baal Gaava? Rav Elyashiv answered him that in fact there is no such thing. "Either you don't know what a Talmid Chochom is or you don't know what a Baal Gaava is." At first glance one would think that Rav Elyashiv is setting a high bar for a Talmid Chochom by declaring that any Baal Gaava is not a Talmid Chochom. Although from hearsay we can't know what Rav Elyashiv meant we can explain this comment with something said by Rav Chaim Volozhin.
Rav Chaim Volozhin in Ruach Chaim says that a Talmid Chochom is vessel that holds Torah. To allow the vessel to hold more content one can shave the inside walls. The thinner the walls the more area for holding Torah. For a Talmid Chochom to acquire more Torah, he needs to minimize himself. He needs to be all Torah will as little of himself as possible. Rav Chaim Volozhin goes on to say that Moshe's towering superiority over everyone else is Torah was just a reflection of his ability to minimize himself more than anyone. If someone were to attain a level of humility on par with Moshe, his level of Torah would also match Moshe's.
The gemara in Sotah (5a) says that Hashem says that He cannot live together in the world with a Baal Gaava. Based on Rav Chaim Volozhin's words the explanation is that a Baal Gaava is a person who does not submit himself to Hashem. He considers himself no ones vessel and lives to serve himself. He therefore does not leave room for Torah or Hashem in his world.
Now we can understand Rav Elyashiv's quote. A Talmid Chochom cannot be a Baal Gaava. Gaava means to think of oneself as independent person meant to serve himself. We do what's good for us, and it may even include learning Torah, Davening and performing Mitzvos. A Talmid Chochom does not live for himself. He has made himself into a mere vessel to hold Hashem and His Torah. Sure his walls may be thicker or thinner, but in principle he has subjugated his being to Hashem. It is us who have chosen to live our lives serving ourselves, even if our life's mission is to do enough mitzvos to go to Gan Eden, who are the real Baalei Gaava.