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Mirrors > Home > MPE Home > Th. List > opelcnv | Structured version Visualization version GIF version |
Description: Ordered-pair membership in converse. (Contributed by NM, 13-Aug-1995.) |
Ref | Expression |
---|---|
opelcnv.1 | ⊢ 𝐴 ∈ V |
opelcnv.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opelcnv | ⊢ (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelcnv.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelcnv.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opelcnvg 5224 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ◡ccnv 5037 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 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-br 4584 df-opab 4644 df-cnv 5046 |
This theorem is referenced by: cnvopab 5452 cnv0OLD 5455 cnvdif 5458 dfrel2 5502 cnvcnvsn 5530 cnvresima 5541 dfco2 5551 cnviin 5589 fcnvres 5995 cnvf1olem 7162 cnvimadfsn 7191 dmtpos 7251 dftpos4 7258 tpostpos 7259 brsdom2 7969 fsumcom2 14347 fsumcom2OLD 14348 fprodcom2 14553 fprodcom2OLD 14554 gsumcom2 18197 metustsym 22170 cnvco1 30903 cnvco2 30904 cnviun 36961 |
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