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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 5721 | . 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 2108 〈cop 4632 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: otelxp1 5730 dff3 7120 ressnop0 7173 swoord1 8777 swoord2 8778 isfin4p1 10355 canthp1lem2 10693 ciclcl 17846 txcmplem1 23649 txlm 23656 dvbsss 25937 nvvcop 30613 nvvop 30628 fldextfld1 33700 prsdm 33913 linedegen 36144 bj-opelresdm 37146 bj-idres 37161 opelopab3 37725 et-ltneverrefl 46886 natglobalincr 46892 fuco1 49016 fuco2 49018 fucoid2 49044 fucocolem2 49049 |
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