Theorem rex0 4314

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

Assertion

Ref	Expression
rex0	⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Proof of Theorem rex0

Step	Hyp	Ref	Expression
1		noel 4292	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3066	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∃wrex 3062  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-dif 3906  df-nul 4288

This theorem is referenced by:  reu0  4315  rmo0  4316  0iun  5020  0qs  8711  sup0riota  9381  cfeq0  10178  cfsuc  10179  hashge2el2difr  14416  cshws0  17041  addsrid  27972  muls01  28120  mulsrid  28121  elons2  28266  onaddscl  28285  onmulscl  28286  n0cut  28342  0ringirng  33866  dya2iocuni  34460  eulerpartlemgh  34555  pmapglb2xN  40142  elpadd0  40179  tfsconcatb0  43695  sprsymrelfvlem  47844  ipolub00  49346