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Mirrors > Home > MPE Home > Th. List > n0moeu | Structured version Visualization version GIF version |
Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010.) |
Ref | Expression |
---|---|
n0moeu | ⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4347 | . . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
2 | 1 | biimpi 215 | . . 3 ⊢ (𝐴 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐴) |
3 | 2 | biantrurd 532 | . 2 ⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴))) |
4 | df-eu 2559 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | bitr4di 289 | 1 ⊢ (𝐴 ≠ ∅ → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1774 ∈ wcel 2099 ∃*wmo 2528 ∃!weu 2558 ≠ wne 2937 ∅c0 4323 |
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-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-eu 2559 df-clab 2706 df-cleq 2720 df-ne 2938 df-dif 3950 df-nul 4324 |
This theorem is referenced by: minveclem4a 25371 |
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