Theorem disjxsn 4112

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 4092	. 2 |- (Disj x e. { A } B <-> A. y E* x ( x e. { A } /\ y e. B ) )
2		moeq 2995	. . 3 |- E* x x = A
3		elsni 3712	. . . . 5 |- ( x e. { A } -> x = A )
4	3	adantr 276	. . . 4 |- ( ( x e. { A } /\ y e. B ) -> x = A )
5	4	moimi 2148	. . 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 1502	1 |- Disj x e. { A } B

Syntax hints:   /\ wa 104   = wceq 1398   E*wmo 2083   e. wcel 2205   {csn 3694  Disj wdisj 4090

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rmo 2530  df-v 2817  df-sn 3700  df-disj 4091

This theorem is referenced by:  disjx0  4113