Theorem disjxsn 3922

Description: A singleton collection is disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)

Assertion

Ref	Expression
disjxsn	|- Disj x e. { A } B

Distinct variable group:   x, A

Allowed substitution hint:   B( x)

Proof of Theorem disjxsn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdisj2 3903	. 2 |- (Disj x e. { A } B <-> A. y E* x ( x e. { A } /\ y e. B ) )
2		moeq 2854	. . 3 |- E* x x = A
3		elsni 3540	. . . . 5 |- ( x e. { A } -> x = A )
4	3	adantr 274	. . . 4 |- ( ( x e. { A } /\ y e. B ) -> x = A )
5	4	moimi 2062	. . 3 |- ( E* x x = A -> E* x ( x e. { A } /\ y e. B ) )
6	2, 5	ax-mp 5	. 2 |- E* x ( x e. { A } /\ y e. B )
7	1, 6	mpgbir 1429	1 |- Disj x e. { A } B

Syntax hints:   /\ wa 103   = wceq 1331   e. wcel 1480   E*wmo 1998   {csn 3522  Disj wdisj 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rmo 2422  df-v 2683  df-sn 3528  df-disj 3902

This theorem is referenced by:  disjx0  3923