Theorem disjexc 28251

Description: A variant of disjex 28250, applicable for more generic families. (Contributed by Thierry Arnoux, 4-Oct-2016.)

Hypothesis

Ref	Expression
disjexc.1	|- ( x = y -> A = B )

Assertion

Ref	Expression
disjexc	|- ( ( E. z ( z e. A /\ z e. B ) -> x = y ) -> ( A = B \/ ( A i^i B ) = (/) ) )

Distinct variable groups:   z, A   z, B

Allowed substitution hints:   A( x, y)   B( x, y)

Proof of Theorem disjexc

Step	Hyp	Ref	Expression
1		disjexc.1	. . 3 |- ( x = y -> A = B )
2	1	imim2i 16	. 2 |- ( ( E. z ( z e. A /\ z e. B ) -> x = y ) -> ( E. z ( z e. A /\ z e. B ) -> A = B ) )
3		orcom 393	. . 3 |- ( ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
4		df-in 3422	. . . . . . 7 |- ( A i^i B ) = { z | ( z e. A /\ z e. B ) }
5	4	neeq1i 2699	. . . . . 6 |- ( ( A i^i B ) =/= (/) <-> { z | ( z e. A /\ z e. B ) } =/= (/) )
6		abn0 3762	. . . . . 6 |- ( { z | ( z e. A /\ z e. B ) } =/= (/) <-> E. z ( z e. A /\ z e. B ) )
7	5, 6	bitr2i 258	. . . . 5 |- ( E. z ( z e. A /\ z e. B ) <-> ( A i^i B ) =/= (/) )
8	7	necon2bbii 2686	. . . 4 |- ( ( A i^i B ) = (/) <-> -. E. z ( z e. A /\ z e. B ) )
9	8	orbi2i 526	. . 3 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) )
10		imor 418	. . 3 |- ( ( E. z ( z e. A /\ z e. B ) -> A = B ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
11	3, 9, 10	3bitr4i 285	. 2 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( E. z ( z e. A /\ z e. B ) -> A = B ) )
12	2, 11	sylibr 217	1 |- ( ( E. z ( z e. A /\ z e. B ) -> x = y ) -> ( A = B \/ ( A i^i B ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 374   /\ wa 375   = wceq 1454   E.wex 1673   e. wcel 1897   {cab 2447   =/= wne 2632   i^i cin 3414   (/)c0 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-in 3422  df-nul 3743

This theorem is referenced by: (None)