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Mirrors > Home > MPE Home > Th. List > disjdif2 | Structured version Visualization version GIF version |
Description: The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.) |
Ref | Expression |
---|---|
disjdif2 | ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq2 4017 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅)) | |
2 | difin 4162 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
3 | dif0 4273 | . 2 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
4 | 1, 2, 3 | 3eqtr3g 2794 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∖ cdif 3850 ∩ cin 3852 ∅c0 4223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-in 3860 df-nul 4224 |
This theorem is referenced by: opwo0id 5365 setsfun0 16701 cnfldfunALT 20330 ptbasfi 22432 undif5 30540 fzdif2 30786 fzodif2 30787 chtvalz 32275 sltlpss 33773 bj-2upln1upl 34900 dvrelog2 39754 dvrelog3 39755 gneispace 41362 dvmptfprodlem 43103 |
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