Theorem sndisj 4644

Description: Any collection of singletons is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
sndisj	|- Disj x e. A { x }

Proof of Theorem sndisj

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 4622	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 3382	. . 3 |- E* x x = y
3		simpr 477	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 4193	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 208	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	equcomd 1946	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 2520	. . 3 |- ( E* x x = y -> E* x ( x e. A /\ y e. { x } ) )
8	2, 7	ax-mp 5	. 2 |- E* x ( x e. A /\ y e. { x } )
9	1, 8	mpgbir 1726	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 384   e. wcel 1990   E*wmo 2471   {csn 4177  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-rmo 2920  df-v 3202  df-sn 4178  df-disj 4621

This theorem is referenced by:  0disj  4645  sibfof  30402  disjsnxp  39239  vonct  40907