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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 3646. (Contributed by BJ, 24-Sep-2022.) |
Ref | Expression |
---|---|
mosneq | ⊢ ∃*𝑥{𝑥} = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqtr3 2766 | . . . 4 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → {𝑥} = {𝑦}) | |
2 | vex 3435 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4777 | . . . 4 ⊢ ({𝑥} = {𝑦} → 𝑥 = 𝑦) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
5 | 4 | gen2 1803 | . 2 ⊢ ∀𝑥∀𝑦(({𝑥} = 𝐴 ∧ {𝑦} = 𝐴) → 𝑥 = 𝑦) |
6 | sneq 4577 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
7 | 6 | eqeq1d 2742 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥} = 𝐴 ↔ {𝑦} = 𝐴)) |
8 | 7 | mo4 2568 | . 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 1540 = wceq 1542 ∃*wmo 2540 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-mo 2542 df-clab 2718 df-cleq 2732 df-clel 2818 df-v 3433 df-sn 4568 |
This theorem is referenced by: pwfir 8939 euabsneu 44488 |
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