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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axsn | Structured version Visualization version GIF version | ||
| Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn 37034). (Contributed by BJ, 12-Jan-2025.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| bj-axsn | ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | velsn 4642 | . 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥) | |
| 2 | 1 | bj-clex 37032 | 1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {csn 4626 |
| 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-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sn 4627 |
| This theorem is referenced by: bj-snexg 37035 |
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