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Mirrors > Home > ILE Home > Th. List > opelvv | GIF version |
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelvv.1 | ⊢ 𝐴 ∈ V |
opelvv.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opelvv | ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelvv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelvv.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opelxpi 4658 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
4 | 1, 2, 3 | mp2an 426 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2148 Vcvv 2737 〈cop 3595 × cxp 4624 |
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-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-opab 4065 df-xp 4632 |
This theorem is referenced by: relsnop 4732 relopabi 4752 eqop2 6178 |
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