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  8713  sup0riota  9383  cfeq0  10180  cfsuc  10181  hashge2el2difr  14418  cshws0  17043  addsrid  27977  muls01  28125  mulsrid  28126  elons2  28271  onaddscl  28290  onmulscl  28291  n0cut  28347  0ringirng  33873  dya2iocuni  34467  eulerpartlemgh  34562  pmapglb2xN  40177  elpadd0  40214  tfsconcatb0  43730  sprsymrelfvlem  47879  ipolub00  49381