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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 4120 | . 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ ∅)) | |
| 2 | difin 4272 | . 2 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
| 3 | dif0 4378 | . 2 ⊢ (𝐴 ∖ ∅) = 𝐴 | |
| 4 | 1, 2, 3 | 3eqtr3g 2800 | 1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (𝐴 ∖ 𝐵) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∖ cdif 3948 ∩ cin 3950 ∅c0 4333 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 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-in 3958 df-nul 4334 | 
| This theorem is referenced by: undif5 4485 opwo0id 5502 setsfun0 17209 cnfldfun 21378 cnfldfunOLD 21391 ptbasfi 23589 sltlpss 27945 slelss 27946 fzdif2 32792 fzodif2 32793 chtvalz 34644 bj-2upln1upl 37025 disjresdif 38243 dvrelog2 42065 dvrelog3 42066 readvrec2 42391 readvrec 42392 gneispace 44147 dvmptfprodlem 45959 | 
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