Theorem sorpssi 6477

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 4773	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C \/ B = C \/ C [ C.] B ) )
2		elex 3087	. . . . . 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 6473	. . . . 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 3087	. . . . . 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 6473	. . . . 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 1289	. . 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 3567	. 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 964   = wceq 1370   e. wcel 1758   _Vcvv 3078   C_ wss 3437   C. wpss 3438   class class class wbr 4401   Or wor 4749   [ C.] crpss 6470

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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640

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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-so 4751  df-xp 4955  df-rel 4956  df-rpss 6471

This theorem is referenced by:  sorpssun  6478  sorpssin  6479  sorpssuni  6480  sorpssint  6481  sorpsscmpl  6482  enfin2i  8602  fin1a2lem9  8689  fin1a2lem10  8690  fin1a2lem11  8691  fin1a2lem13  8693