Theorem rexn0 3567

Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3568). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Assertion

Ref	Expression
rexn0	|- ( E. x e. A ph -> A =/= (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rexn0

Step	Hyp	Ref	Expression
1		ne0i 3475	. . 3 |- ( x e. A -> A =/= (/) )
2	1	a1d 22	. 2 |- ( x e. A -> ( ph -> A =/= (/) ) )
3	2	rexlimiv 2619	1 |- ( E. x e. A ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2178   =/= wne 2378   E.wrex 2487   (/)c0 3468

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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-nul 3469

This theorem is referenced by: (None)