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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 6163 | . 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴) | |
| 2 | relcnv 6122 | . . 3 ⊢ Rel ◡𝐴 | |
| 3 | relnonrel 43600 | . . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅) | |
| 4 | 2, 3 | mpbi 230 | . 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅ |
| 5 | 1, 4 | eqtri 2765 | 1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∅c0 4333 ◡ccnv 5684 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: brnonrel 43602 dmnonrel 43603 resnonrel 43605 cononrel1 43607 cononrel2 43608 clcnvlem 43636 cnvrcl0 43638 |
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