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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 1500 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | eu3h 2045 | . . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))) |
4 | 3 | simplbi2com 1421 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥𝜑 → ∃!𝑥𝜑)) |
5 | df-mo 2004 | . 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 1330 Ⅎwnf 1437 ∃wex 1469 ∃!weu 2000 ∃*wmo 2001 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 |
This theorem is referenced by: mo2icl 2867 rmo2ilem 3002 dffun5r 5143 frecuzrdgtcl 10216 frecuzrdgfunlem 10223 |
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