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Mirrors > Home > MPE Home > Th. List > difun1 | Structured version Visualization version GIF version |
Description: A relationship involving double difference and union. (Contributed by NM, 29-Aug-2004.) |
Ref | Expression |
---|---|
difun1 | ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4048 | . . . 4 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) | |
2 | invdif 4098 | . . . 4 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) | |
3 | 1, 2 | eqtr3i 2851 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) |
4 | undm 4115 | . . . . 5 ⊢ (V ∖ (𝐵 ∪ 𝐶)) = ((V ∖ 𝐵) ∩ (V ∖ 𝐶)) | |
5 | 4 | ineq2i 4038 | . . . 4 ⊢ (𝐴 ∩ (V ∖ (𝐵 ∪ 𝐶))) = (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) |
6 | invdif 4098 | . . . 4 ⊢ (𝐴 ∩ (V ∖ (𝐵 ∪ 𝐶))) = (𝐴 ∖ (𝐵 ∪ 𝐶)) | |
7 | 5, 6 | eqtr3i 2851 | . . 3 ⊢ (𝐴 ∩ ((V ∖ 𝐵) ∩ (V ∖ 𝐶))) = (𝐴 ∖ (𝐵 ∪ 𝐶)) |
8 | 3, 7 | eqtr3i 2851 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = (𝐴 ∖ (𝐵 ∪ 𝐶)) |
9 | invdif 4098 | . . 3 ⊢ (𝐴 ∩ (V ∖ 𝐵)) = (𝐴 ∖ 𝐵) | |
10 | 9 | difeq1i 3951 | . 2 ⊢ ((𝐴 ∩ (V ∖ 𝐵)) ∖ 𝐶) = ((𝐴 ∖ 𝐵) ∖ 𝐶) |
11 | 8, 10 | eqtr3i 2851 | 1 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 Vcvv 3414 ∖ cdif 3795 ∪ cun 3796 ∩ cin 3797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 |
This theorem is referenced by: dif32 4120 difabs 4121 difpr 4552 infdiffi 8832 mreexexlem4d 16660 nulmbl2 23702 unmbl 23703 caragenuncllem 41520 |
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