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Mirrors > Home > ILE Home > Th. List > iindif2m | GIF version |
Description: Indexed intersection of class difference. Compare to Theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.) |
Ref | Expression |
---|---|
iindif2m | ⊢ (∃x x ∈ A → ∩ x ∈ A (B ∖ 𝐶) = (B ∖ ∪ x ∈ A 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.28mv 3308 | . . . 4 ⊢ (∃x x ∈ A → (∀x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶) ↔ (y ∈ B ∧ ∀x ∈ A ¬ y ∈ 𝐶))) | |
2 | eldif 2921 | . . . . . 6 ⊢ (y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ 𝐶)) | |
3 | 2 | bicomi 123 | . . . . 5 ⊢ ((y ∈ B ∧ ¬ y ∈ 𝐶) ↔ y ∈ (B ∖ 𝐶)) |
4 | 3 | ralbii 2324 | . . . 4 ⊢ (∀x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶) ↔ ∀x ∈ A y ∈ (B ∖ 𝐶)) |
5 | ralnex 2310 | . . . . . 6 ⊢ (∀x ∈ A ¬ y ∈ 𝐶 ↔ ¬ ∃x ∈ A y ∈ 𝐶) | |
6 | eliun 3652 | . . . . . 6 ⊢ (y ∈ ∪ x ∈ A 𝐶 ↔ ∃x ∈ A y ∈ 𝐶) | |
7 | 5, 6 | xchbinxr 607 | . . . . 5 ⊢ (∀x ∈ A ¬ y ∈ 𝐶 ↔ ¬ y ∈ ∪ x ∈ A 𝐶) |
8 | 7 | anbi2i 430 | . . . 4 ⊢ ((y ∈ B ∧ ∀x ∈ A ¬ y ∈ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∪ x ∈ A 𝐶)) |
9 | 1, 4, 8 | 3bitr3g 211 | . . 3 ⊢ (∃x x ∈ A → (∀x ∈ A y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∪ x ∈ A 𝐶))) |
10 | vex 2554 | . . . 4 ⊢ y ∈ V | |
11 | eliin 3653 | . . . 4 ⊢ (y ∈ V → (y ∈ ∩ x ∈ A (B ∖ 𝐶) ↔ ∀x ∈ A y ∈ (B ∖ 𝐶))) | |
12 | 10, 11 | ax-mp 7 | . . 3 ⊢ (y ∈ ∩ x ∈ A (B ∖ 𝐶) ↔ ∀x ∈ A y ∈ (B ∖ 𝐶)) |
13 | eldif 2921 | . . 3 ⊢ (y ∈ (B ∖ ∪ x ∈ A 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ ∪ x ∈ A 𝐶)) | |
14 | 9, 12, 13 | 3bitr4g 212 | . 2 ⊢ (∃x x ∈ A → (y ∈ ∩ x ∈ A (B ∖ 𝐶) ↔ y ∈ (B ∖ ∪ x ∈ A 𝐶))) |
15 | 14 | eqrdv 2035 | 1 ⊢ (∃x x ∈ A → ∩ x ∈ A (B ∖ 𝐶) = (B ∖ ∪ x ∈ A 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 ∃wrex 2301 Vcvv 2551 ∖ cdif 2908 ∪ ciun 3648 ∩ ciin 3649 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-dif 2914 df-iun 3650 df-iin 3651 |
This theorem is referenced by: (None) |
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