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| 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.) | 
| Ref | Expression | 
|---|---|
| bj-nul | ⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | 
| 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) | 
| Colors of variables: wff setvar class | 
| 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) | 
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