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Mirrors > Home > MPE Home > Th. List > moel | Structured version Visualization version GIF version |
Description: "At most one" element in a set. (Contributed by Thierry Arnoux, 26-Jul-2018.) Avoid ax-11 2147. (Revised by Wolf Lammen, 23-Nov-2024.) |
Ref | Expression |
---|---|
moel | ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2812 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | 1 | mo4 2556 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) |
3 | r2al 3191 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑥 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 ∈ wcel 2099 ∃*wmo 2528 ∀wral 3058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-mo 2530 df-clel 2806 df-ral 3059 |
This theorem is referenced by: disjnf 32359 isthinc3 48029 isthincd2lem1 48033 mndtcbas2 48095 |
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