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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 5667 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simprbi 498 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 〈cop 4591 × cxp 5629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-opab 5167 df-xp 5637 |
This theorem is referenced by: dff4 7046 eceqoveq 8695 axdc4lem 10325 canthp1lem2 10523 cicrcl 17622 txcmplem1 22920 txlm 22927 brcgr 27654 nvex 29358 fldextfld2 32129 prsrn 32276 pprodss4v 34400 poimirlem27 36036 natglobalincr 44907 |
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