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Theorem bj-nul 36568
Description: Two formulations of the axiom of the empty set ax-nul 5310. Proposal: place it right before ax-nul 5310. (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 3486 . 2 (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2 eq0 4347 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32exbii 1842 . 2 (∃𝑥 𝑥 = ∅ ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
41, 3bitri 274 1 (∅ ∈ V ↔ ∃𝑥𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1531   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  c0 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3475  df-dif 3952  df-nul 4327
This theorem is referenced by: (None)
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