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Theorem bj-nul 33860
 Description: Two formulations of the axiom of the empty set ax-nul 5068. Proposal: place it right before ax-nul 5068. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nul (∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nul
StepHypRef Expression
1 isset 3427 . 2 (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2 eq0 4196 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32exbii 1810 . 2 (∃𝑥 𝑥 = ∅ ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
41, 3bitri 267 1 (∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1505   = wceq 1507  ∃wex 1742   ∈ wcel 2050  Vcvv 3415  ∅c0 4180 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-11 2093  ax-12 2106  ax-ext 2750 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-dif 3834  df-nul 4181 This theorem is referenced by: (None)
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