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Mirrors > Home > MPE Home > Th. List > opelvv | Structured version Visualization version GIF version |
Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by NM, 22-Aug-2013.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opelvv.1 | ⊢ 𝐴 ∈ V |
opelvv.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opelvv | ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelvv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelvv.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opelxpi 5573 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2112 Vcvv 3398 〈cop 4533 × cxp 5534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 df-xp 5542 |
This theorem is referenced by: relopabiALT 5678 funsneqopb 6945 isof1oopb 7112 1st2ndb 7779 eqop2 7782 evlfcl 17684 brtxp 33868 brpprod 33873 brsset 33877 brcart 33920 brcup 33927 brcap 33928 elcnvlem 40826 |
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