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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 5591 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 Vcvv 3494 〈cop 4572 × cxp 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-opab 5128 df-xp 5560 |
This theorem is referenced by: relopabiALT 5694 funsneqopb 6913 isof1oopb 7077 1st2ndb 7728 eqop2 7731 evlfcl 17471 brtxp 33341 brpprod 33346 brsset 33350 brcart 33393 brcup 33400 brcap 33401 elcnvlem 39959 |
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