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Theorem bj-nul 32209
 Description: Two formulations of the axiom of the empty set ax-nul 4717. Proposal: place it right before ax-nul 4717. (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 3180 . 2 (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2 eq0 3888 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32exbii 1764 . 2 (∃𝑥 𝑥 = ∅ ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
41, 3bitri 263 1 (∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by: (None)
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