Theorem rexn0 3522

Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3523). (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 3430	. . 3 |- ( x e. A -> A =/= (/) )
2	1	a1d 22	. 2 |- ( x e. A -> ( ph -> A =/= (/) ) )
3	2	rexlimiv 2588	1 |- ( E. x e. A ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2148   =/= wne 2347   E.wrex 2456   (/)c0 3423

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-nul 3424

This theorem is referenced by: (None)