Theorem bj-nul 37057

Description: Two formulations of the axiom of the empty set ax-nul 5306. Proposal: place it right before ax-nul 5306. (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 3494	. 2 ⊢ (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2		eq0 4350	. . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3	2	exbii 1848	. 2 ⊢ (∃𝑥 𝑥 = ∅ ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	1, 3	bitri 275	1 ⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-nul 4334

This theorem is referenced by: (None)