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Mirrors > Home > ILE Home > Th. List > mosubop | GIF version |
Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
mosubop.1 | ⊢ ∃*𝑥𝜑 |
Ref | Expression |
---|---|
mosubop | ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mosubop.1 | . . 3 ⊢ ∃*𝑥𝜑 | |
2 | 1 | gen2 1339 | . 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑 |
3 | mosubopt 4405 | . 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑)) | |
4 | 2, 3 | ax-mp 7 | 1 ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∀wal 1241 = wceq 1243 ∃wex 1381 ∃*wmo 1901 〈cop 3378 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 |
This theorem is referenced by: ovi3 5637 ov6g 5638 oprabex3 5756 |
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