Theorem rex0 4360

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 4338	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3074	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2108  ∃wrex 3070  ∅c0 4333

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-dif 3954  df-nul 4334

This theorem is referenced by:  reu0  4361  rmo0  4362  0iun  5063  0qs  8807  sup0riota  9505  cfeq0  10296  cfsuc  10297  hashge2el2difr  14520  cshws0  17139  addsrid  27997  muls01  28138  mulsrid  28139  elons2  28281  onaddscl  28286  onmulscl  28287  n0scut  28338  1p1e2s  28400  0ringirng  33739  dya2iocuni  34285  eulerpartlemgh  34380  pmapglb2xN  39774  elpadd0  39811  tfsconcatb0  43357  sprsymrelfvlem  47477  ipolub00  48882