![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iinvdif | Structured version Visualization version GIF version |
Description: The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.) |
Ref | Expression |
---|---|
iinvdif | ⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4384 | . . . 4 ⊢ (V ∖ ∅) = V | |
2 | 0iun 5068 | . . . . 5 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | difeq2i 4133 | . . . 4 ⊢ (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅) |
4 | 0iin 5069 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = V | |
5 | 1, 3, 4 | 3eqtr4ri 2774 | . . 3 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) |
6 | iineq1 5014 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = ∩ 𝑥 ∈ ∅ (V ∖ 𝐵)) | |
7 | iuneq1 5013 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵) | |
8 | 7 | difeq2d 4136 | . . 3 ⊢ (𝐴 = ∅ → (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵)) |
9 | 5, 6, 8 | 3eqtr4a 2801 | . 2 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) |
10 | iindif2 5082 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
11 | 9, 10 | pm2.61ine 3023 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 ∖ cdif 3960 ∅c0 4339 ∪ ciun 4996 ∩ ciin 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 df-iun 4998 df-iin 4999 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |