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Mirrors > Home > MPE Home > Th. List > exmoeu | Structured version Visualization version GIF version |
Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.) |
Ref | Expression |
---|---|
exmoeu | ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mo 2463 | . . . 4 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
2 | 1 | biimpi 205 | . . 3 ⊢ (∃*𝑥𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
3 | 2 | com12 32 | . 2 ⊢ (∃𝑥𝜑 → (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
4 | exmo 2483 | . . . . 5 ⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑) | |
5 | 4 | ori 389 | . . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑) |
6 | 5 | con1i 143 | . . 3 ⊢ (¬ ∃*𝑥𝜑 → ∃𝑥𝜑) |
7 | euex 2482 | . . 3 ⊢ (∃!𝑥𝜑 → ∃𝑥𝜑) | |
8 | 6, 7 | ja 172 | . 2 ⊢ ((∃*𝑥𝜑 → ∃!𝑥𝜑) → ∃𝑥𝜑) |
9 | 3, 8 | impbii 198 | 1 ⊢ (∃𝑥𝜑 ↔ (∃*𝑥𝜑 → ∃!𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∃wex 1695 ∃!weu 2458 ∃*wmo 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-eu 2462 df-mo 2463 |
This theorem is referenced by: (None) |
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