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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnvnonrel | Structured version Visualization version GIF version |
Description: The converse of the non-relation part of a class is empty. (Contributed by RP, 18-Oct-2020.) |
Ref | Expression |
---|---|
cnvnonrel | ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvdif 6166 | . 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴) | |
2 | relcnv 6125 | . . 3 ⊢ Rel ◡𝐴 | |
3 | relnonrel 43577 | . . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅) | |
4 | 2, 3 | mpbi 230 | . 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅ |
5 | 1, 4 | eqtri 2763 | 1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3960 ∅c0 4339 ◡ccnv 5688 Rel wrel 5694 |
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-in 3970 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-xp 5695 df-rel 5696 df-cnv 5697 |
This theorem is referenced by: brnonrel 43579 dmnonrel 43580 resnonrel 43582 cononrel1 43584 cononrel2 43585 clcnvlem 43613 cnvrcl0 43615 |
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