Theorem sorpssin 6611

Description: A chain of sets is closed under binary intersection. (Contributed by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
sorpssin	|- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B i^i C ) e. A )

Proof of Theorem sorpssin

Step	Hyp	Ref	Expression
1		simprl 769	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. A )
2		df-ss 3430	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
3		eleq1 2528	. . . 4 |- ( ( B i^i C ) = B -> ( ( B i^i C ) e. A <-> B e. A ) )
4	2, 3	sylbi 200	. . 3 |- ( B C_ C -> ( ( B i^i C ) e. A <-> B e. A ) )
5	1, 4	syl5ibrcom 230	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C -> ( B i^i C ) e. A ) )
6		simprr 771	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. A )
7		dfss1 3649	. . . 4 |- ( C C_ B <-> ( B i^i C ) = C )
8		eleq1 2528	. . . 4 |- ( ( B i^i C ) = C -> ( ( B i^i C ) e. A <-> C e. A ) )
9	7, 8	sylbi 200	. . 3 |- ( C C_ B -> ( ( B i^i C ) e. A <-> C e. A ) )
10	6, 9	syl5ibrcom 230	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( C C_ B -> ( B i^i C ) e. A ) )
11		sorpssi 6609	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )
12	5, 10, 11	mpjaod 387	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B i^i C ) e. A )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   i^i cin 3415   C_ wss 3416   Or wor 4776   [ C.] crpss 6602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-so 4778  df-xp 4862  df-rel 4863  df-rpss 6603

This theorem is referenced by: (None)