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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 5713 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) | |
2 | 1 | simplbi 499 | 1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) → 𝐴 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ⟨cop 4635 × cxp 5675 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 df-xp 5683 |
This theorem is referenced by: otelxp1 5722 dff3 7102 ressnop0 7151 swoord1 8734 swoord2 8735 isfin4p1 10310 canthp1lem2 10648 ciclcl 17749 txcmplem1 23145 txlm 23152 dvbsss 25419 nvvcop 29847 nvvop 29862 fldextfld1 32728 prsdm 32894 linedegen 35115 bj-opelresdm 36026 bj-idres 36041 opelopab3 36586 et-ltneverrefl 45587 natglobalincr 45591 |
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