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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 5721 | . 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 2107 Vcvv 3479 〈cop 4631 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-opab 5205 df-xp 5690 | 
| This theorem is referenced by: relopabiALT 5832 funsneqopb 7171 isof1oopb 7346 1st2ndb 8055 eqop2 8058 evlfcl 18268 brtxp 35882 brpprod 35887 brsset 35891 brcart 35934 brcup 35941 brcap 35942 elcnvlem 43619 swapfelvv 48987 fucoelvv 49038 | 
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