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Mirrors > Home > ILE Home > Th. List > iundif2ss | GIF version |
Description: Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.) |
Ref | Expression |
---|---|
iundif2ss | ⊢ ∪ x ∈ A (B ∖ 𝐶) ⊆ (B ∖ ∩ x ∈ A 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 2921 | . . . . . 6 ⊢ (y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ¬ y ∈ 𝐶)) | |
2 | 1 | rexbii 2325 | . . . . 5 ⊢ (∃x ∈ A y ∈ (B ∖ 𝐶) ↔ ∃x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶)) |
3 | r19.42v 2461 | . . . . 5 ⊢ (∃x ∈ A (y ∈ B ∧ ¬ y ∈ 𝐶) ↔ (y ∈ B ∧ ∃x ∈ A ¬ y ∈ 𝐶)) | |
4 | 2, 3 | bitri 173 | . . . 4 ⊢ (∃x ∈ A y ∈ (B ∖ 𝐶) ↔ (y ∈ B ∧ ∃x ∈ A ¬ y ∈ 𝐶)) |
5 | rexnalim 2311 | . . . . . 6 ⊢ (∃x ∈ A ¬ y ∈ 𝐶 → ¬ ∀x ∈ A y ∈ 𝐶) | |
6 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
7 | eliin 3653 | . . . . . . 7 ⊢ (y ∈ V → (y ∈ ∩ x ∈ A 𝐶 ↔ ∀x ∈ A y ∈ 𝐶)) | |
8 | 6, 7 | ax-mp 7 | . . . . . 6 ⊢ (y ∈ ∩ x ∈ A 𝐶 ↔ ∀x ∈ A y ∈ 𝐶) |
9 | 5, 8 | sylnibr 601 | . . . . 5 ⊢ (∃x ∈ A ¬ y ∈ 𝐶 → ¬ y ∈ ∩ x ∈ A 𝐶) |
10 | 9 | anim2i 324 | . . . 4 ⊢ ((y ∈ B ∧ ∃x ∈ A ¬ y ∈ 𝐶) → (y ∈ B ∧ ¬ y ∈ ∩ x ∈ A 𝐶)) |
11 | 4, 10 | sylbi 114 | . . 3 ⊢ (∃x ∈ A y ∈ (B ∖ 𝐶) → (y ∈ B ∧ ¬ y ∈ ∩ x ∈ A 𝐶)) |
12 | eliun 3652 | . . 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 | 11, 12, 13 | 3imtr4i 190 | . 2 ⊢ (y ∈ ∪ x ∈ A (B ∖ 𝐶) → y ∈ (B ∖ ∩ x ∈ A 𝐶)) |
15 | 14 | ssriv 2943 | 1 ⊢ ∪ x ∈ A (B ∖ 𝐶) ⊆ (B ∖ ∩ x ∈ A 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ↔ wb 98 ∈ wcel 1390 ∀wral 2300 ∃wrex 2301 Vcvv 2551 ∖ cdif 2908 ⊆ wss 2911 ∪ 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-in 2918 df-ss 2925 df-iun 3650 df-iin 3651 |
This theorem is referenced by: (None) |
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