Theorem sorpssun 6944

Description: A chain of sets is closed under binary union. (Contributed by Mario Carneiro, 16-May-2015.)

Assertion

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

Proof of Theorem sorpssun

Step	Hyp	Ref	Expression
1		simprr 796	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> C e. A )
2		ssequn1 3783	. . . 4 |- ( B C_ C <-> ( B u. C ) = C )
3		eleq1 2689	. . . 4 |- ( ( B u. C ) = C -> ( ( B u. C ) e. A <-> C e. A ) )
4	2, 3	sylbi 207	. . 3 |- ( B C_ C -> ( ( B u. C ) e. A <-> C e. A ) )
5	1, 4	syl5ibrcom 237	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C -> ( B u. C ) e. A ) )
6		simprl 794	. . 3 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> B e. A )
7		ssequn2 3786	. . . 4 |- ( C C_ B <-> ( B u. C ) = B )
8		eleq1 2689	. . . 4 |- ( ( B u. C ) = B -> ( ( B u. C ) e. A <-> B e. A ) )
9	7, 8	sylbi 207	. . 3 |- ( C C_ B -> ( ( B u. C ) e. A <-> B e. A ) )
10	6, 9	syl5ibrcom 237	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( C C_ B -> ( B u. C ) e. A ) )
11		sorpssi 6943	. 2 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B C_ C \/ C C_ B ) )
12	5, 10, 11	mpjaod 396	1 |- ( ( [ C.] Or A /\ ( B e. A /\ C e. A ) ) -> ( B u. C ) e. A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   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:  finsschain  8273  lbsextlem2  19159  lbsextlem3  19160  filssufilg  21715