Theorem rex0 4312

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 4290	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3064	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2113  ∃wrex 3060  ∅c0 4285

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-dif 3904  df-nul 4286

This theorem is referenced by:  reu0  4313  rmo0  4314  0iun  5018  0qs  8700  sup0riota  9369  cfeq0  10166  cfsuc  10167  hashge2el2difr  14404  cshws0  17029  addsrid  27960  muls01  28108  mulsrid  28109  elons2  28254  onaddscl  28273  onmulscl  28274  n0cut  28330  0ringirng  33846  dya2iocuni  34440  eulerpartlemgh  34535  pmapglb2xN  40028  elpadd0  40065  tfsconcatb0  43582  sprsymrelfvlem  47732  ipolub00  49234