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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 5072 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-xp 5044 |
This theorem is referenced by: relsnop 5147 relopabiALT 5168 funsneqop 6323 isof1oopb 6475 1st2ndb 7097 eqop2 7100 evlfcl 16685 vtxvalsnop 25716 iedgvalsnop 25717 brtxp 31157 brpprod 31162 brsset 31166 brcart 31209 brcup 31216 brcap 31217 elcnvlem 36926 |
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