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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nul | Structured version Visualization version GIF version |
Description: Two formulations of the axiom of the empty set ax-nul 5068. Proposal: place it right before ax-nul 5068. (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 3427 | . 2 ⊢ (∅ ∈ V ↔ ∃𝑥 𝑥 = ∅) | |
2 | eq0 4196 | . . 3 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
3 | 2 | exbii 1810 | . 2 ⊢ (∃𝑥 𝑥 = ∅ ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
4 | 1, 3 | bitri 267 | 1 ⊢ (∅ ∈ V ↔ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∀wal 1505 = wceq 1507 ∃wex 1742 ∈ wcel 2050 Vcvv 3415 ∅c0 4180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-v 3417 df-dif 3834 df-nul 4181 |
This theorem is referenced by: (None) |
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