Theorem disjnf 29384

Description: In case x is not free in B, disjointness is not so interesting since it reduces to cases where A is a singleton. (Google Groups discussion with Peter Mazsa.) (Contributed by Thierry Arnoux, 26-Jul-2018.)

Assertion

Ref	Expression
disjnf	|- (Disj x e. A B <-> ( B = (/) \/ E* x x e. A ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem disjnf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inidm 3822	. . . 4 |- ( B i^i B ) = B
2	1	eqeq1i 2627	. . 3 |- ( ( B i^i B ) = (/) <-> B = (/) )
3	2	orbi1i 542	. 2 |- ( ( ( B i^i B ) = (/) \/ A. x e. A A. y e. A x = y ) <-> ( B = (/) \/ A. x e. A A. y e. A x = y ) )
4		eqidd 2623	. . . 4 |- ( x = y -> B = B )
5	4	disjor 4634	. . 3 |- (Disj x e. A B <-> A. x e. A A. y e. A ( x = y \/ ( B i^i B ) = (/) ) )
6		orcom 402	. . . . . 6 |- ( ( x = y \/ ( B i^i B ) = (/) ) <-> ( ( B i^i B ) = (/) \/ x = y ) )
7	6	ralbii 2980	. . . . 5 |- ( A. y e. A ( x = y \/ ( B i^i B ) = (/) ) <-> A. y e. A ( ( B i^i B ) = (/) \/ x = y ) )
8		r19.32v 3083	. . . . 5 |- ( A. y e. A ( ( B i^i B ) = (/) \/ x = y ) <-> ( ( B i^i B ) = (/) \/ A. y e. A x = y ) )
9	7, 8	bitri 264	. . . 4 |- ( A. y e. A ( x = y \/ ( B i^i B ) = (/) ) <-> ( ( B i^i B ) = (/) \/ A. y e. A x = y ) )
10	9	ralbii 2980	. . 3 |- ( A. x e. A A. y e. A ( x = y \/ ( B i^i B ) = (/) ) <-> A. x e. A ( ( B i^i B ) = (/) \/ A. y e. A x = y ) )
11		r19.32v 3083	. . 3 |- ( A. x e. A ( ( B i^i B ) = (/) \/ A. y e. A x = y ) <-> ( ( B i^i B ) = (/) \/ A. x e. A A. y e. A x = y ) )
12	5, 10, 11	3bitri 286	. 2 |- (Disj x e. A B <-> ( ( B i^i B ) = (/) \/ A. x e. A A. y e. A x = y ) )
13		moel 29323	. . 3 |- ( E* x x e. A <-> A. x e. A A. y e. A x = y )
14	13	orbi2i 541	. 2 |- ( ( B = (/) \/ E* x x e. A ) <-> ( B = (/) \/ A. x e. A A. y e. A x = y ) )
15	3, 12, 14	3bitr4i 292	1 |- (Disj x e. A B <-> ( B = (/) \/ E* x x e. A ) )

Syntax hints:   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   E*wmo 2471   A.wral 2912   i^i cin 3573   (/)c0 3915  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-nfc 2753  df-ral 2917  df-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by: (None)