Theorem sorpssi 6568

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 4823	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C \/ B = C \/ C [ C.] B ) )
2		elex 3122	. . . . . 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 6564	. . . . 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 3122	. . . . . 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 6564	. . . . 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 3606	. 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 1379   e. wcel 1767   _Vcvv 3113   C_ wss 3476   C. wpss 3477   class class class wbr 4447   Or wor 4799   [ C.] crpss 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-so 4801  df-xp 5005  df-rel 5006  df-rpss 6562

This theorem is referenced by:  sorpssun  6569  sorpssin  6570  sorpssuni  6571  sorpssint  6572  sorpsscmpl  6573  enfin2i  8697  fin1a2lem9  8784  fin1a2lem10  8785  fin1a2lem11  8786  fin1a2lem13  8788