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Theorem bj-nul 34473
 Description: Two formulations of the axiom of the empty set ax-nul 5174. Proposal: place it right before ax-nul 5174. (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 3453 . 2 (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2 eq0 4258 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32exbii 1849 . 2 (∃𝑥 𝑥 = ∅ ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
41, 3bitri 278 1 (∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  ∅c0 4243 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-nul 4244 This theorem is referenced by: (None)
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