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Mirrors > Home > ILE Home > Th. List > mo2r | GIF version |
Description: A condition which implies "at most one." (Contributed by Jim Kingdon, 2-Jul-2018.) |
Ref | Expression |
---|---|
mo2r.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
mo2r | ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mo2r.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1484 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | eu3h 2022 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
4 | 3 | simplbi2com 1405 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | df-mo 1981 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
6 | 4, 5 | sylibr 133 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1314 Ⅎwnf 1421 ∃wex 1453 ∃!weu 1977 ∃*wmo 1978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 |
This theorem is referenced by: mo2icl 2836 rmo2ilem 2970 dffun5r 5105 frecuzrdgtcl 10153 frecuzrdgfunlem 10160 |
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