Theorem rex0 4357

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 4330	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3074	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2106  ∃wrex 3070  ∅c0 4322

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-dif 3951  df-nul 4323

This theorem is referenced by:  reu0  4358  rmo0  4359  0iun  5066  sup0riota  9459  cfeq0  10250  cfsuc  10251  hashge2el2difr  14441  cshws0  17034  addsrid  27445  muls01  27565  mulsrid  27566  0ringirng  32748  dya2iocuni  33277  eulerpartlemgh  33372  0qs  37234  pmapglb2xN  38638  elpadd0  38675  tfsconcatb0  42084  sprsymrelfvlem  46148  ipolub00  47608