![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4280 | . 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) | |
2 | invdif 4267 | . . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
3 | 2 | eqcomi 2735 | . . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶)) |
4 | 3 | difeq2i 4115 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) |
5 | dfin2 4259 | . . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶)) | |
6 | 5 | uneq2i 4157 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) |
7 | 1, 4, 6 | 3eqtr4i 2764 | 1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 Vcvv 3462 ∖ cdif 3943 ∪ cun 3944 ∩ cin 3945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 |
This theorem is referenced by: psdmullem 22155 restmetu 24567 difelcarsg 34157 mblfinlem3 37373 mblfinlem4 37374 |
Copyright terms: Public domain | W3C validator |