Theorem sorpssin 6561

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 754	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. A )
2		df-ss 3475	. . . 4 |- ( B C_ C <-> ( B i^i C ) = B )
3		eleq1 2526	. . . 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 755	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. A )
7		dfss1 3689	. . . 4 |- ( C C_ B <-> ( B i^i C ) = C )
8		eleq1 2526	. . . 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 6559	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )
12	5, 10, 11	mpjaod 379	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B i^i C ) e. A )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   i^i cin 3460   C_ wss 3461   Or wor 4788   [ C.] crpss 6552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-so 4790  df-xp 4994  df-rel 4995  df-rpss 6553

This theorem is referenced by: (None)