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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 6002 | . 2 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = (◡𝐴 ∖ ◡◡◡𝐴) | |
2 | relcnv 5967 | . . 3 ⊢ Rel ◡𝐴 | |
3 | relnonrel 39967 | . . 3 ⊢ (Rel ◡𝐴 ↔ (◡𝐴 ∖ ◡◡◡𝐴) = ∅) | |
4 | 2, 3 | mpbi 232 | . 2 ⊢ (◡𝐴 ∖ ◡◡◡𝐴) = ∅ |
5 | 1, 4 | eqtri 2844 | 1 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∖ cdif 3933 ∅c0 4291 ◡ccnv 5554 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 |
This theorem is referenced by: brnonrel 39969 dmnonrel 39970 resnonrel 39972 cononrel1 39974 cononrel2 39975 clcnvlem 40003 cnvrcl0 40005 |
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