Theorem sorpssin 6471

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 755	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. A )
2		df-ss 3443	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
3		eleq1 2523	. . . 4 |- ( ( B i^i C ) = B -> ( ( B i^i C ) e. A <-> B e. A ) )
4	2, 3	sylbi 195	. . 3 |- ( B C_ C -> ( ( B i^i C ) e. A <-> B e. A ) )
5	1, 4	syl5ibrcom 222	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C -> ( B i^i C ) e. A ) )
6		simprr 756	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. A )
7		dfss1 3656	. . . 4 |- ( C C_ B <-> ( B i^i C ) = C )
8		eleq1 2523	. . . 4 |- ( ( B i^i C ) = C -> ( ( B i^i C ) e. A <-> C e. A ) )
9	7, 8	sylbi 195	. . 3 |- ( C C_ B -> ( ( B i^i C ) e. A <-> C e. A ) )
10	6, 9	syl5ibrcom 222	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( C C_ B -> ( B i^i C ) e. A ) )
11		sorpssi 6469	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )
12	5, 10, 11	mpjaod 381	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B i^i C ) e. A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   i^i cin 3428   C_ wss 3429   Or wor 4741   [ C.] crpss 6462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-so 4743  df-xp 4947  df-rel 4948  df-rpss 6463

This theorem is referenced by: (None)