Theorem sndisj 4107

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 4089	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 2994	. . 3 |- E* x x = y
3		simpr 110	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 3708	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 122	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	eqcomd 2240	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 2148	. . 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 1502	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 104   E*wmo 2083   e. wcel 2205   {csn 3691  Disj wdisj 4087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rmo 2530  df-v 2817  df-sn 3697  df-disj 4088

This theorem is referenced by:  0disj  4108  disjsnxp  6435