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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 5714 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simprbi 495 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 〈cop 4636 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5212 df-xp 5684 |
This theorem is referenced by: dff4 7110 eceqoveq 8841 axdc4lem 10480 canthp1lem2 10678 cicrcl 17789 txcmplem1 23589 txlm 23596 brcgr 28783 nvex 30493 fldextfld2 33473 prsrn 33647 pprodss4v 35611 poimirlem27 37251 natglobalincr 46401 |
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