Theorem sorpssi 6603

Description: Property of a chain of sets. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
sorpssi	|- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )

Proof of Theorem sorpssi

Step	Hyp	Ref	Expression
1		solin 4796	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C \/ B = C \/ C [ C.] B ) )
2		elex 3065	. . . . . 6 |- ( C e. A -> C e. _V )
3	2	ad2antll 740	. . . . 5 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. _V )
4		brrpssg 6599	. . . . 5 |- ( C e. _V -> ( B [ C.] C <-> B C. C ) )
5	3, 4	syl 17	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C <-> B C. C ) )
6		biidd 245	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> B = C ) )
7		elex 3065	. . . . . 6 |- ( B e. A -> B e. _V )
8	7	ad2antrl 739	. . . . 5 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. _V )
9		brrpssg 6599	. . . . 5 |- ( B e. _V -> ( C [ C.] B <-> C C. B ) )
10	8, 9	syl 17	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( C [ C.] B <-> C C. B ) )
11	5, 6, 10	3orbi123d 1347	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( ( B [ C.] C \/ B = C \/ C [ C.] B ) <-> ( B C. C \/ B = C \/ C C. B ) ) )
12	1, 11	mpbid 215	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C. C \/ B = C \/ C C. B ) )
13		sspsstri 3546	. 2 |- ( ( B C_ C \/ C C_ B ) <-> ( B C. C \/ B = C \/ C C. B ) )
14	12, 13	sylibr 217	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   \/ w3o 990   = wceq 1454   e. wcel 1897   _Vcvv 3056   C_ wss 3415   C. wpss 3416   class class class wbr 4415   Or wor 4772   [ C.] crpss 6596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-so 4774  df-xp 4858  df-rel 4859  df-rpss 6597

This theorem is referenced by:  sorpssun  6604  sorpssin  6605  sorpssuni  6606  sorpssint  6607  sorpsscmpl  6608  enfin2i  8776  fin1a2lem9  8863  fin1a2lem10  8864  fin1a2lem11  8865  fin1a2lem13  8867