Theorem ndisj2 39218

Description: A non disjointness condition. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
ndisj2.1	|- ( x = y -> B = C )

Assertion

Ref	Expression
ndisj2	|- ( -. Disj x e. A B <-> E. x e. A E. y e. A ( x =/= y /\ ( B i^i C ) =/= (/) ) )

Distinct variable groups:   x, A, y   y, B   x, C

Allowed substitution hints:   B( x)   C( y)

Proof of Theorem ndisj2

Step	Hyp	Ref	Expression
1		ndisj2.1	. . . 4 |- ( x = y -> B = C )
2	1	disjor 4634	. . 3 |- (Disj x e. A B <-> A. x e. A A. y e. A ( x = y \/ ( B i^i C ) = (/) ) )
3	2	notbii 310	. 2 |- ( -. Disj x e. A B <-> -. A. x e. A A. y e. A ( x = y \/ ( B i^i C ) = (/) ) )
4		rexnal 2995	. 2 |- ( E. x e. A -. A. y e. A ( x = y \/ ( B i^i C ) = (/) ) <-> -. A. x e. A A. y e. A ( x = y \/ ( B i^i C ) = (/) ) )
5		rexnal 2995	. . . 4 |- ( E. y e. A -. ( x = y \/ ( B i^i C ) = (/) ) <-> -. A. y e. A ( x = y \/ ( B i^i C ) = (/) ) )
6		ioran 511	. . . . . 6 |- ( -. ( x = y \/ ( B i^i C ) = (/) ) <-> ( -. x = y /\ -. ( B i^i C ) = (/) ) )
7		df-ne 2795	. . . . . . 7 |- ( x =/= y <-> -. x = y )
8		df-ne 2795	. . . . . . 7 |- ( ( B i^i C ) =/= (/) <-> -. ( B i^i C ) = (/) )
9	7, 8	anbi12i 733	. . . . . 6 |- ( ( x =/= y /\ ( B i^i C ) =/= (/) ) <-> ( -. x = y /\ -. ( B i^i C ) = (/) ) )
10	6, 9	bitr4i 267	. . . . 5 |- ( -. ( x = y \/ ( B i^i C ) = (/) ) <-> ( x =/= y /\ ( B i^i C ) =/= (/) ) )
11	10	rexbii 3041	. . . 4 |- ( E. y e. A -. ( x = y \/ ( B i^i C ) = (/) ) <-> E. y e. A ( x =/= y /\ ( B i^i C ) =/= (/) ) )
12	5, 11	bitr3i 266	. . 3 |- ( -. A. y e. A ( x = y \/ ( B i^i C ) = (/) ) <-> E. y e. A ( x =/= y /\ ( B i^i C ) =/= (/) ) )
13	12	rexbii 3041	. 2 |- ( E. x e. A -. A. y e. A ( x = y \/ ( B i^i C ) = (/) ) <-> E. x e. A E. y e. A ( x =/= y /\ ( B i^i C ) =/= (/) ) )
14	3, 4, 13	3bitr2i 288	1 |- ( -. Disj x e. A B <-> E. x e. A E. y e. A ( x =/= y /\ ( B i^i C ) =/= (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   A.wral 2912   E.wrex 2913   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-ne 2795  df-ral 2917  df-rex 2918  df-rmo 2920  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916  df-disj 4621

This theorem is referenced by:  disjrnmpt2  39375