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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 5148 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ (V × V)) | |
4 | 1, 2, 3 | mp2an 708 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 Vcvv 3200 〈cop 4183 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 |
This theorem is referenced by: relsnop 5224 relopabiALT 5246 funsneqop 6418 isof1oopb 6575 1st2ndb 7206 eqop2 7209 evlfcl 16862 brtxp 31987 brpprod 31992 brsset 31996 brcart 32039 brcup 32046 brcap 32047 elcnvlem 37907 |
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