Theorem sorpssi 6943

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 5058	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B [ C.] C \/ B = C \/ C [ C.] B ) )
2		elex 3212	. . . . . 6 |- ( C e. A -> C e. _V )
3	2	ad2antll 765	. . . . 5 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. _V )
4		brrpssg 6939	. . . . 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 252	. . . 4 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B = C <-> B = C ) )
7		elex 3212	. . . . . 6 |- ( B e. A -> B e. _V )
8	7	ad2antrl 764	. . . . 5 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. _V )
9		brrpssg 6939	. . . . 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 1398	. . 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 222	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C. C \/ B = C \/ C C. B ) )
13		sspsstri 3709	. 2 |- ( ( B C_ C \/ C C_ B ) <-> ( B C. C \/ B = C \/ C C. B ) )
14	12, 13	sylibr 224	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   C. wpss 3575   class class class wbr 4653   Or wor 5034   [ C.] crpss 6936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-so 5036  df-xp 5120  df-rel 5121  df-rpss 6937

This theorem is referenced by:  sorpssun  6944  sorpssin  6945  sorpssuni  6946  sorpssint  6947  sorpsscmpl  6948  enfin2i  9143  fin1a2lem9  9230  fin1a2lem10  9231  fin1a2lem11  9232  fin1a2lem13  9234