Theorem sndisj 4029

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 4012	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 2939	. . 3 |- E* x x = y
3		simpr 110	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 3639	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 122	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	eqcomd 2202	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 2110	. . 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 1467	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 104   E*wmo 2046   e. wcel 2167   {csn 3622  Disj wdisj 4010

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rmo 2483  df-v 2765  df-sn 3628  df-disj 4011

This theorem is referenced by:  0disj  4030  disjsnxp  6295