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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 5724 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simprbi 496 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 〈cop 4636 × cxp 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-xp 5694 |
This theorem is referenced by: dff4 7120 eceqoveq 8860 axdc4lem 10492 canthp1lem2 10690 cicrcl 17850 txcmplem1 23664 txlm 23671 brcgr 28929 nvex 30639 fldextfld2 33677 prsrn 33875 pprodss4v 35865 poimirlem27 37633 natglobalincr 46830 |
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