Theorem bj-nul 37720

Description: Two formulations of the axiom of the empty set ax-nul 5269. Proposal: place it right before ax-nul 5269. (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 3469	. 2 ⊢ (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅)
2		eq0 4304	. . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥)
3	2	exbii 1878	. 2 ⊢ (∃𝑥 𝑥 = ∅ ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
4	1, 3	bitri 278	1 ⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ∅c0 4286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-nul 4287

This theorem is used by: (None)