Theorem rex0 3757

Description: Vacuous existential quantification is false. (Contributed by NM, 15-Oct-2003.)

Assertion

Ref	Expression
rex0	|- -. E. x e. (/) ph

Proof of Theorem rex0

Step	Hyp	Ref	Expression
1		noel 3746	. . 3 |- -. x e. (/)
2	1	pm2.21i 136	. 2 |- ( x e. (/) -> -. ph )
3	2	nrex 2853	1 |- -. E. x e. (/) ph

Syntax hints:   -. wn 3   e. wcel 1897   E.wrex 2749   (/)c0 3742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-rex 2754  df-v 3058  df-dif 3418  df-nul 3743

This theorem is referenced by:  0iun  4348  sup0riota  8004  cfeq0  8711  cfsuc  8712  hashge2el2difr  12670  cshws0  15120  meet0  16431  join0  16432  dya2iocuni  29153  eulerpartlemgh  29259  pmapglb2xN  33381  elpadd0  33418