Theorem disjxsn 4031

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 4012	. 2 |- (Disj x e. { A } B <-> A. y E* x ( x e. { A } /\ y e. B ) )
2		moeq 2939	. . 3 |- E* x x = A
3		elsni 3640	. . . . 5 |- ( x e. { A } -> x = A )
4	3	adantr 276	. . . 4 |- ( ( x e. { A } /\ y e. B ) -> x = A )
5	4	moimi 2110	. . 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 1467	1 |- Disj x e. { A } B

Syntax hints:   /\ wa 104   = wceq 1364   E*wmo 2046   e. wcel 2167   {csn 3622  Disj wdisj 4010

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rmo 2483  df-v 2765  df-sn 3628  df-disj 4011

This theorem is referenced by:  disjx0  4032