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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 3642. (Contributed by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
mosneq | ⊢ ∃*𝑥{𝑥} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2764 | . . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦}) | |
2 | vex 3436 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4771 | . . . 4 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
5 | 4 | gen2 1799 | . 2 ⊢ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
6 | sneq 4571 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 6 | eqeq1d 2740 | . . 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 230 | 1 ⊢ ∃*𝑥{𝑥} = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1537 = wceq 1539 ∃*wmo 2538 {csn 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1542 df-ex 1783 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-v 3434 df-sn 4562 |
This theorem is referenced by: pwfir 8959 euabsneu 44522 |
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