Theorem rexn0 2346

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)

Assertion

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

Distinct variable group:   x,A

Proof of Theorem rexn0

Step	Hyp	Ref	Expression
1		ne0i 2276	. . 3 |- (x e. A -> A =/= (/))
2	1	a1d 12	. 2 |- (x e. A -> (ph -> A =/= (/)))
3	2	r19.23aiv 1735	1 |- (E.x e. A ph -> A =/= (/))

Syntax hints:   -> wi 3   e. wcel 955   =/= wne 1577  E.wrex 1638  (/)c0 2270

This theorem is referenced by:  grpn0 7980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-nul 2271

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