Theorem rexn0 3319

Description: Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3320). (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 3230	. . 3 |- ( x e. A -> A =/= (/) )
2	1	a1d 22	. 2 |- ( x e. A -> ( ph -> A =/= (/) ) )
3	2	rexlimiv 2427	1 |- ( E. x e. A ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1393   =/= wne 2204   E.wrex 2307   (/)c0 3224

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-nul 3225

This theorem is referenced by: (None)