![]() |
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 4364 | . . . 4 ⊢ (V ∖ ∅) = V | |
2 | 0iun 5056 | . . . . 5 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | difeq2i 4111 | . . . 4 ⊢ (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅) |
4 | 0iin 5057 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = V | |
5 | 1, 3, 4 | 3eqtr4ri 2763 | . . 3 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) |
6 | iineq1 5004 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = ∩ 𝑥 ∈ ∅ (V ∖ 𝐵)) | |
7 | iuneq1 5003 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵) | |
8 | 7 | difeq2d 4114 | . . 3 ⊢ (𝐴 = ∅ → (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵)) |
9 | 5, 6, 8 | 3eqtr4a 2790 | . 2 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) |
10 | iindif2 5070 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
11 | 9, 10 | pm2.61ine 3017 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3466 ∖ cdif 3937 ∅c0 4314 ∪ ciun 4987 ∩ ciin 4988 |
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-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-in 3947 df-ss 3957 df-nul 4315 df-iun 4989 df-iin 4990 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |