Theorem rex0 4323

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 4301	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3057	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2109  ∃wrex 3053  ∅c0 4296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-dif 3917  df-nul 4297

This theorem is referenced by:  reu0  4324  rmo0  4325  0iun  5027  0qs  8736  sup0riota  9417  cfeq0  10209  cfsuc  10210  hashge2el2difr  14446  cshws0  17072  addsrid  27871  muls01  28015  mulsrid  28016  elons2  28159  onaddscl  28174  onmulscl  28175  n0scut  28226  1p1e2s  28302  0ringirng  33684  dya2iocuni  34274  eulerpartlemgh  34369  pmapglb2xN  39766  elpadd0  39803  tfsconcatb0  43333  sprsymrelfvlem  47491  ipolub00  48981