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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 3904 | . . . 4 ⊢ (V ∖ ∅) = V | |
2 | 0iun 4513 | . . . . 5 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | difeq2i 3687 | . . . 4 ⊢ (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅) |
4 | 0iin 4514 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = V | |
5 | 1, 3, 4 | 3eqtr4ri 2643 | . . 3 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) |
6 | iineq1 4471 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = ∩ 𝑥 ∈ ∅ (V ∖ 𝐵)) | |
7 | iuneq1 4470 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵) | |
8 | 7 | difeq2d 3690 | . . 3 ⊢ (𝐴 = ∅ → (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵)) |
9 | 5, 6, 8 | 3eqtr4a 2670 | . 2 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) |
10 | iindif2 4525 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
11 | 9, 10 | pm2.61ine 2865 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 ∪ ciun 4455 ∩ ciin 4456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-iun 4457 df-iin 4458 |
This theorem is referenced by: (None) |
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