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 ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nul

Step	Hyp	Ref	Expression
1		isset 3486	. 2 ⊢ (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2		eq0 4347	. . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3	2	exbii 1842	. 2 ⊢ (∃𝑥 𝑥 = ∅ ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	1, 3	bitri 274	1 ⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

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)