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Mirrors > Home > MPE Home > Th. List > opelcnv | Structured version Visualization version GIF version |
Description: Ordered-pair membership in converse relation. (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 5894 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅)) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∈ wcel 2106 Vcvv 3478 〈cop 4637 ◡ccnv 5688 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-cnv 5697 |
This theorem is referenced by: cnvopab 6160 cnvopabOLD 6161 cnvdif 6166 dfrel2 6211 cnvcnvsn 6241 cnvresima 6252 dfco2 6267 cnviin 6308 fcnvres 6786 cnvf1olem 8134 cnvimadfsn 8196 dmtpos 8262 dftpos4 8269 tpostpos 8270 brsdom2 9136 fsumcom2 15807 fprodcom2 16017 gsumcom2 20008 metustsym 24584 gsumhashmul 33047 cnvco1 35739 cnvco2 35740 cnviun 43640 |
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