Theorem rex0 3794

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 3784	. . 3 |- -. x e. (/)
2	1	pm2.21i 131	. 2 |- ( x e. (/) -> -. ph )
3	2	nrex 2914	1 |- -. E. x e. (/) ph

Syntax hints:   -. wn 3   e. wcel 1762   E.wrex 2810   (/)c0 3780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-dif 3474  df-nul 3781

This theorem is referenced by:  0iun  4377  cfeq0  8627  cfsuc  8628  cshws0  14435  meet0  15615  join0  15616  dya2iocuni  27882  eulerpartlemgh  27945  pmapglb2xN  34445  elpadd0  34482