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Mirrors > Home > ILE Home > Th. List > rmo2i | GIF version |
Description: Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.) |
Ref | Expression |
---|---|
rmo2.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
rmo2i | ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexex 2410 | . 2 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦)) | |
2 | rmo2.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | 2 | rmo2ilem 2903 | . 2 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) |
4 | 1, 3 | syl 14 | 1 ⊢ (∃𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝑦) → ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 Ⅎwnf 1389 ∃wex 1421 ∀wral 2348 ∃wrex 2349 ∃*wrmo 2351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-ral 2353 df-rex 2354 df-rmo 2356 |
This theorem is referenced by: (None) |
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