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Mirrors > Home > MPE Home > Th. List > Mathboxes > nonrel | Structured version Visualization version GIF version |
Description: A non-relation is equal to the base class with all ordered pairs removed. (Contributed by RP, 25-Oct-2020.) |
Ref | Expression |
---|---|
nonrel | ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvcnv 6016 | . . 3 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V)) | |
2 | 1 | difeq2i 4047 | . 2 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (𝐴 ∩ (V × V))) |
3 | difin 4188 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ (V × V))) = (𝐴 ∖ (V × V)) | |
4 | 2, 3 | eqtri 2821 | 1 ⊢ (𝐴 ∖ ◡◡𝐴) = (𝐴 ∖ (V × V)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 × cxp 5517 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 |
This theorem is referenced by: elnonrel 40285 |
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