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Mirrors > Home > MPE Home > Th. List > difdif2 | Structured version Visualization version GIF version |
Description: Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.) |
Ref | Expression |
---|---|
difdif2 | ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difindi 4282 | . 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) | |
2 | invdif 4269 | . . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
3 | 2 | eqcomi 2736 | . . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶)) |
4 | 3 | difeq2i 4117 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
5 | dfin2 4261 | . . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶)) | |
6 | 5 | uneq2i 4159 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) |
7 | 1, 4, 6 | 3eqtr4i 2765 | 1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3471 ∖ cdif 3944 ∪ cun 3945 ∩ cin 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 |
This theorem is referenced by: psdmullem 22094 restmetu 24497 difelcarsg 33935 mblfinlem3 37137 mblfinlem4 37138 |
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