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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 5584 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simprbi 499 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐵 ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 〈cop 4565 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-opab 5120 df-xp 5554 |
This theorem is referenced by: dff4 6860 eceqoveq 8394 axdc4lem 9869 canthp1lem2 10067 cicrcl 17065 txcmplem1 22241 txlm 22248 brcgr 26678 nvex 28380 fldextfld2 31033 prsrn 31151 pprodss4v 33338 poimirlem27 34911 |
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