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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 1499 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | eu3h 2051 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
4 | 3 | simplbi2com 1424 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | df-mo 2010 | . 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 1333 Ⅎwnf 1440 ∃wex 1472 ∃!weu 2006 ∃*wmo 2007 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 |
This theorem is referenced by: mo2icl 2891 rmo2ilem 3026 dffun5r 5185 frecuzrdgtcl 10321 frecuzrdgfunlem 10328 |
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