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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 1450 | . 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑 |
3 | mosubopt 4691 | . 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑)) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = 〈𝑦, 𝑧〉 ∧ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∀wal 1351 = wceq 1353 ∃wex 1492 ∃*wmo 2027 〈cop 3595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 |
This theorem is referenced by: ovi3 6010 ov6g 6011 oprabex3 6129 axaddf 7866 axmulf 7867 |
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