Theorem rex0 3938

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 3919	. . 3 |- -. x e. (/)
2	1	pm2.21i 116	. 2 |- ( x e. (/) -> -. ph )
3	2	nrex 3000	1 |- -. E. x e. (/) ph

Syntax hints:   -. wn 3   e. wcel 1990   E.wrex 2913   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  0iun  4577  sup0riota  8371  cfeq0  9078  cfsuc  9079  hashge2el2difr  13263  cshws0  15808  meet0  17137  join0  17138  dya2iocuni  30345  eulerpartlemgh  30440  0qs  34133  pmapglb2xN  35058  elpadd0  35095  sprsymrelfvlem  41740