Theorem rex0 4358

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 4331	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3075	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2107  ∃wrex 3071  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-dif 3952  df-nul 4324

This theorem is referenced by:  reu0  4359  rmo0  4360  0iun  5067  sup0riota  9460  cfeq0  10251  cfsuc  10252  hashge2el2difr  14442  cshws0  17035  addsrid  27448  muls01  27568  mulsrid  27569  0ringirng  32753  dya2iocuni  33282  eulerpartlemgh  33377  0qs  37239  pmapglb2xN  38643  elpadd0  38680  tfsconcatb0  42094  sprsymrelfvlem  46158  ipolub00  47618