Theorem sndisj 4040

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 4023	. 2 |- (Disj x e. A { x } <-> A. y E* x ( x e. A /\ y e. { x } ) )
2		moeq 2948	. . 3 |- E* x x = y
3		simpr 110	. . . . . 6 |- ( ( x e. A /\ y e. { x } ) -> y e. { x } )
4		velsn 3650	. . . . . 6 |- ( y e. { x } <-> y = x )
5	3, 4	sylib 122	. . . . 5 |- ( ( x e. A /\ y e. { x } ) -> y = x )
6	5	eqcomd 2211	. . . 4 |- ( ( x e. A /\ y e. { x } ) -> x = y )
7	6	moimi 2119	. . 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 1476	1 |- Disj x e. A { x }

Syntax hints:   /\ wa 104   E*wmo 2055   e. wcel 2176   {csn 3633  Disj wdisj 4021

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rmo 2492  df-v 2774  df-sn 3639  df-disj 4022

This theorem is referenced by:  0disj  4041  disjsnxp  6323