Theorem disjeq1 4627

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjeq1	|- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem disjeq1

Step	Hyp	Ref	Expression
1		eqimss2 3658	. . 3 |- ( A = B -> B C_ A )
2		disjss1 4626	. . 3 |- ( B C_ A -> (Disj x e. A C -> Disj x e. B C ) )
3	1, 2	syl 17	. 2 |- ( A = B -> (Disj x e. A C -> Disj x e. B C ) )
4		eqimss 3657	. . 3 |- ( A = B -> A C_ B )
5		disjss1 4626	. . 3 |- ( A C_ B -> (Disj x e. B C -> Disj x e. A C ) )
6	4, 5	syl 17	. 2 |- ( A = B -> (Disj x e. B C -> Disj x e. A C ) )
7	3, 6	impbid 202	1 |- ( A = B -> (Disj x e. A C <-> Disj x e. B C ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   C_ wss 3574  Disj wdisj 4620

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rmo 2920  df-in 3581  df-ss 3588  df-disj 4621

This theorem is referenced by:  disjeq1d  4628  volfiniun  23315  disjrnmpt  29398  iundisj2cnt  29558  unelldsys  30221  sigapildsys  30225  ldgenpisyslem1  30226  rossros  30243  measvun  30272  pmeasmono  30386  pmeasadd  30387  meadjuni  40674