Theorem rex0 4317

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 4296	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3269	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2114  ∃wrex 3139  ∅c0 4291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ral 3143  df-rex 3144  df-dif 3939  df-nul 4292

This theorem is referenced by:  reu0  4318  rmo0  4319  0iun  4986  sup0riota  8929  cfeq0  9678  cfsuc  9679  hashge2el2difr  13840  cshws0  16435  dya2iocuni  31541  eulerpartlemgh  31636  0qs  35637  pmapglb2xN  36923  elpadd0  36960  sprsymrelfvlem  43672