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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 4378 | . . . 4 ⊢ (V ∖ ∅) = V | |
| 2 | 0iun 5063 | . . . . 5 ⊢ ∪ 𝑥 ∈ ∅ 𝐵 = ∅ | |
| 3 | 2 | difeq2i 4123 | . . . 4 ⊢ (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅) |
| 4 | 0iin 5064 | . . . 4 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = V | |
| 5 | 1, 3, 4 | 3eqtr4ri 2776 | . . 3 ⊢ ∩ 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵) |
| 6 | iineq1 5009 | . . 3 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = ∩ 𝑥 ∈ ∅ (V ∖ 𝐵)) | |
| 7 | iuneq1 5008 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ ∅ 𝐵) | |
| 8 | 7 | difeq2d 4126 | . . 3 ⊢ (𝐴 = ∅ → (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) = (V ∖ ∪ 𝑥 ∈ ∅ 𝐵)) |
| 9 | 5, 6, 8 | 3eqtr4a 2803 | . 2 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) |
| 10 | iindif2 5077 | . 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
| 11 | 9, 10 | pm2.61ine 3025 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 (V ∖ 𝐵) = (V ∖ ∪ 𝑥 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 ∪ ciun 4991 ∩ ciin 4992 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-iun 4993 df-iin 4994 |
| This theorem is referenced by: (None) |
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