Theorem disjxsn 4016

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 3997	. 2 |- (Disj x e. { A } B <-> A. y E* x ( x e. { A } /\ y e. B ) )
2		moeq 2927	. . 3 |- E* x x = A
3		elsni 3625	. . . . 5 |- ( x e. { A } -> x = A )
4	3	adantr 276	. . . 4 |- ( ( x e. { A } /\ y e. B ) -> x = A )
5	4	moimi 2103	. . 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 1464	1 |- Disj x e. { A } B

Syntax hints:   /\ wa 104   = wceq 1364   E*wmo 2039   e. wcel 2160   {csn 3607  Disj wdisj 3995

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rmo 2476  df-v 2754  df-sn 3613  df-disj 3996

This theorem is referenced by:  disjx0  4017