Theorem sndisj 3920

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 3903	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 2854	. . 3 |- E* x x = y
3		simpr 109	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 3539	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 121	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	eqcomd 2143	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 2062	. . 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 1429	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 103   e. wcel 1480   E*wmo 1998   {csn 3522  Disj wdisj 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rmo 2422  df-v 2683  df-sn 3528  df-disj 3902

This theorem is referenced by:  0disj  3921  disjsnxp  6127