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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 2156. (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 2823 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 2 | 1 | mo4 2565 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → 𝑥 = 𝑦)) | 
| 3 | r2al 3194 | . 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 206 ∧ wa 395 ∀wal 1537 ∈ wcel 2107 ∃*wmo 2537 ∀wral 3060 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-clel 2815 df-ral 3061 | 
| This theorem is referenced by: disjnf 32584 oppcmndclem 48920 isthinc3 49095 isthincd2lem1 49099 termcbasmo 49154 arweuthinc 49187 arweutermc 49188 mndtcbas2 49235 | 
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