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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 37015). (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 4646 | . 2 ⊢ (𝑧 ∈ {𝑥} ↔ 𝑧 = 𝑥) | |
2 | 1 | bj-clex 37013 | 1 ⊢ ({𝑥} ∈ V ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1534 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 {csn 4630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-v 3479 df-sn 4631 |
This theorem is referenced by: bj-snexg 37016 |
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