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Mirrors > Home > MPE Home > Th. List > opelxp2 | Structured version Visualization version GIF version |
Description: The second member of an ordered pair of classes in a Cartesian product belongs to second Cartesian product argument. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelxp2 | ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5572 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simprbi 500 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 〈cop 4533 × cxp 5534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 df-xp 5542 |
This theorem is referenced by: dff4 6898 eceqoveq 8482 axdc4lem 10034 canthp1lem2 10232 cicrcl 17262 txcmplem1 22492 txlm 22499 brcgr 26945 nvex 28646 fldextfld2 31393 prsrn 31533 pprodss4v 33872 poimirlem27 35490 |
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