Theorem rex0 3798

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 3787	. . 3 |- -. x e. (/)
2	1	pm2.21i 131	. 2 |- ( x e. (/) -> -. ph )
3	2	nrex 2909	1 |- -. E. x e. (/) ph

Syntax hints:   -. wn 3   e. wcel 1823   E.wrex 2805   (/)c0 3783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-nul 3784

This theorem is referenced by:  0iun  4372  cfeq0  8627  cfsuc  8628  cshws0  14673  meet0  15969  join0  15970  dya2iocuni  28494  eulerpartlemgh  28584  pmapglb2xN  35912  elpadd0  35949