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Mirrors > Home > MPE Home > Th. List > mosneq | Structured version Visualization version GIF version |
Description: There exists at most one set whose singleton is equal to a given class. See also moeq 3716. (Contributed by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
mosneq | ⊢ ∃*𝑥{𝑥} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2761 | . . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦}) | |
2 | vex 3482 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4845 | . . . 4 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
5 | 4 | gen2 1793 | . 2 ⊢ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
6 | sneq 4641 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 6 | eqeq1d 2737 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴)) |
8 | 7 | mo4 2564 | . 2 ⊢ (∃*𝑥{𝑥} = 𝐴 ↔ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦)) |
9 | 5, 8 | mpbir 231 | 1 ⊢ ∃*𝑥{𝑥} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃*wmo 2536 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-sn 4632 |
This theorem is referenced by: pwfir 9353 euabsneu 46978 |
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