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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 5712 | . 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 4634 × cxp 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-opab 5211 df-xp 5682 |
This theorem is referenced by: otelxp1 5721 dff3 7101 ressnop0 7153 swoord1 8737 swoord2 8738 isfin4p1 10313 canthp1lem2 10651 ciclcl 17754 txcmplem1 23366 txlm 23373 dvbsss 25652 nvvcop 30115 nvvop 30130 fldextfld1 33017 prsdm 33193 linedegen 35420 bj-opelresdm 36330 bj-idres 36345 opelopab3 36890 et-ltneverrefl 45886 natglobalincr 45890 |
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