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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 4095 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅)) | |
2 | difin 4240 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
3 | dif0 4334 | . 2 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
4 | 1, 2, 3 | 3eqtr3g 2881 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∖ cdif 3935 ∩ cin 3937 ∅c0 4293 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 |
This theorem is referenced by: opwo0id 5389 setsfun0 16521 cnfldfunALT 20560 ptbasfi 22191 fzdif2 30516 fzodif2 30517 chtvalz 31902 bj-2upln1upl 34338 gneispace 40491 dvmptfprodlem 42236 |
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