Theorem rex0 4082

Description: Vacuous 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 4063	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 116	. 2 ⊢ (𝑥 ∈ ∅ → ¬ 𝜑)
3	2	nrex 3139	1 ⊢ ¬ ∃𝑥 ∈ ∅ 𝜑

Syntax hints:  ¬ wn 3   ∈ wcel 2140  ∃wrex 3052  ∅c0 4059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-v 3343  df-dif 3719  df-nul 4060

This theorem is referenced by:  0iun  4730  sup0riota  8539  cfeq0  9291  cfsuc  9292  hashge2el2difr  13476  cshws0  16031  meet0  17359  join0  17360  dya2iocuni  30676  eulerpartlemgh  30771  0qs  34474  pmapglb2xN  35580  elpadd0  35617  sprsymrelfvlem  42269