Theorem disjexc 23986

Description: A variant of disjex 23985, 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 14	. 2 |- ( ( E. z ( z e. A /\ z e. B ) -> x = y ) -> ( E. z ( z e. A /\ z e. B ) -> A = B ) )
3		orcom 377	. . 3 |- ( ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) <-> ( -. E. z ( z e. A /\ z e. B ) \/ A = B ) )
4		df-in 3287	. . . . . . 7 |- ( A i^i B ) = { z | ( z e. A /\ z e. B ) }
5	4	neeq1i 2577	. . . . . 6 |- ( ( A i^i B ) =/= (/) <-> { z | ( z e. A /\ z e. B ) } =/= (/) )
6		abn0 3606	. . . . . 6 |- ( { z | ( z e. A /\ z e. B ) } =/= (/) <-> E. z ( z e. A /\ z e. B ) )
7	5, 6	bitr2i 242	. . . . 5 |- ( E. z ( z e. A /\ z e. B ) <-> ( A i^i B ) =/= (/) )
8	7	necon2bbii 2623	. . . 4 |- ( ( A i^i B ) = (/) <-> -. E. z ( z e. A /\ z e. B ) )
9	8	orbi2i 506	. . 3 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( A = B \/ -. E. z ( z e. A /\ z e. B ) ) )
10		imor 402	. . 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 269	. 2 |- ( ( A = B \/ ( A i^i B ) = (/) ) <-> ( E. z ( z e. A /\ z e. B ) -> A = B ) )
12	2, 11	sylibr 204	1 |- ( ( E. z ( z e. A /\ z e. B ) -> x = y ) -> ( A = B \/ ( A i^i B ) = (/) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 358   /\ wa 359   E.wex 1547   = wceq 1649   e. wcel 1721   {cab 2390   =/= wne 2567   i^i cin 3279   (/)c0 3588

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-in 3287  df-nul 3589