Theorem sndisj 3760

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 3747	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 2716	. . 3 |- E* x x = y
3		simpr 103	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 3392	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 127	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	eqcomd 2045	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 1965	. . 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 1342	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 97   e. wcel 1393   E*wmo 1901   {csn 3375  Disj wdisj 3745

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rmo 2314  df-v 2559  df-sn 3381  df-disj 3746

This theorem is referenced by:  0disj  3761