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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 3713. (Contributed by BJ, 24-Sep-2022.) |
| Ref | Expression |
|---|---|
| mosneq | ⊢ ∃*𝑥{𝑥} = 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqtr3 2763 | . . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦}) | |
| 2 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 3 | 2 | sneqr 4840 | . . . 4 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
| 4 | 1, 3 | syl 17 | . . 3 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
| 5 | 4 | gen2 1796 | . 2 ⊢ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
| 6 | sneq 4636 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
| 7 | 6 | eqeq1d 2739 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴)) |
| 8 | 7 | mo4 2566 | . 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 1538 = wceq 1540 ∃*wmo 2538 {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-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sn 4627 |
| This theorem is referenced by: pwfir 9355 euabsneu 47040 |
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