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Mirrors > Home > MPE Home > Th. List > opelxp1 | Structured version Visualization version GIF version |
Description: The first member of an ordered pair of classes in a Cartesian product belongs to first Cartesian product argument. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelxp1 | ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5724 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simplbi 497 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶) |
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: otelxp1 5733 dff3 7119 ressnop0 7172 swoord1 8775 swoord2 8776 isfin4p1 10352 canthp1lem2 10690 ciclcl 17849 txcmplem1 23664 txlm 23671 dvbsss 25951 nvvcop 30622 nvvop 30637 fldextfld1 33676 prsdm 33874 linedegen 36124 bj-opelresdm 37127 bj-idres 37142 opelopab3 37704 et-ltneverrefl 46826 natglobalincr 46830 |
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