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Mirrors > Home > ILE Home > Th. List > opth2 | GIF version |
Description: Ordered pair theorem. (Contributed by NM, 21-Sep-2014.) |
Ref | Expression |
---|---|
opth2.1 | ⊢ 𝐶 ∈ V |
opth2.2 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
opth2 | ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth2.1 | . 2 ⊢ 𝐶 ∈ V | |
2 | opth2.2 | . 2 ⊢ 𝐷 ∈ V | |
3 | opthg2 4239 | . 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈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-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: eqvinop 4243 opelxp 4656 fsn 5688 dfplpq2 7352 ltresr 7837 frecuzrdgtcl 10411 frecuzrdgfunlem 10418 |
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