Theorem rex0 4366

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 4344	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3072	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2106  ∃wrex 3068  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-dif 3966  df-nul 4340

This theorem is referenced by:  reu0  4367  rmo0  4368  0iun  5068  0qs  8806  sup0riota  9503  cfeq0  10294  cfsuc  10295  hashge2el2difr  14517  cshws0  17136  addsrid  28012  muls01  28153  mulsrid  28154  elons2  28296  onaddscl  28301  onmulscl  28302  n0scut  28353  1p1e2s  28415  0ringirng  33704  dya2iocuni  34265  eulerpartlemgh  34360  pmapglb2xN  39755  elpadd0  39792  tfsconcatb0  43334  sprsymrelfvlem  47415  ipolub00  48782