Theorem sndisj 3871

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 3854	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 2812	. . 3 |- E* x x = y
3		simpr 109	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 3491	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 121	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	eqcomd 2105	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 2025	. . 3 |- ( E* x x = y -> E* x ( x e. A /\ y e. { x } ) )
8	2, 7	ax-mp 7	. 2 |- E* x ( x e. A /\ y e. { x } )
9	1, 8	mpgbir 1397	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 103   e. wcel 1448   E*wmo 1961   {csn 3474  Disj wdisj 3852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rmo 2383  df-v 2643  df-sn 3480  df-disj 3853

This theorem is referenced by:  0disj  3872  disjsnxp  6064