Theorem disjxsn 3980

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 3961	. 2 |- (Disj x e. { A } B <-> A. y E* x ( x e. { A } /\ y e. B ) )
2		moeq 2901	. . 3 |- E* x x = A
3		elsni 3594	. . . . 5 |- ( x e. { A } -> x = A )
4	3	adantr 274	. . . 4 |- ( ( x e. { A } /\ y e. B ) -> x = A )
5	4	moimi 2079	. . 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 1441	1 |- Disj x e. { A } B

Syntax hints:   /\ wa 103   = wceq 1343   E*wmo 2015   e. wcel 2136   {csn 3576  Disj wdisj 3959

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rmo 2452  df-v 2728  df-sn 3582  df-disj 3960

This theorem is referenced by:  disjx0  3981