Theorem sorpssi 6585

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 4832	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C \/ B = C \/ C [ C.] B ) )
2		elex 3118	. . . . . 6 |- ( C e. A -> C e. _V )
3	2	ad2antll 728	. . . . 5 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. _V )
4		brrpssg 6581	. . . . 5 |- ( C e. _V -> ( B [ C.] C <-> B C. C ) )
5	3, 4	syl 16	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C <-> B C. C ) )
6		biidd 237	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> B = C ) )
7		elex 3118	. . . . . 6 |- ( B e. A -> B e. _V )
8	7	ad2antrl 727	. . . . 5 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. _V )
9		brrpssg 6581	. . . . 5 |- ( B e. _V -> ( C [ C.] B <-> C C. B ) )
10	8, 9	syl 16	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( C [ C.] B <-> C C. B ) )
11	5, 6, 10	3orbi123d 1298	. . 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 210	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C. C \/ B = C \/ C C. B ) )
13		sspsstri 3602	. 2 |- ( ( B C_ C \/ C C_ B ) <-> ( B C. C \/ B = C \/ C C. B ) )
14	12, 13	sylibr 212	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   \/ w3o 972   = wceq 1395   e. wcel 1819   _Vcvv 3109   C_ wss 3471   C. wpss 3472   class class class wbr 4456   Or wor 4808   [ C.] crpss 6578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-so 4810  df-xp 5014  df-rel 5015  df-rpss 6579

This theorem is referenced by:  sorpssun  6586  sorpssin  6587  sorpssuni  6588  sorpssint  6589  sorpsscmpl  6590  enfin2i  8718  fin1a2lem9  8805  fin1a2lem10  8806  fin1a2lem11  8807  fin1a2lem13  8809