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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 2982 | . . . . . 6 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶)) | |
2 | 1 | rexbii 2373 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶)) |
3 | r19.42v 2511 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶)) | |
4 | 2, 3 | bitri 182 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶)) |
5 | rexnalim 2359 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 → ¬ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶) | |
6 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
7 | eliin 3683 | . . . . . . 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 634 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶 → ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) |
10 | 9 | anim2i 334 | . . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ¬ 𝑦 ∈ 𝐶) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)) |
11 | 4, 10 | sylbi 119 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶) → (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)) |
12 | eliun 3682 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∖ 𝐶)) | |
13 | eldif 2982 | . . 3 ⊢ (𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶)) | |
14 | 11, 12, 13 | 3imtr4i 199 | . 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) → 𝑦 ∈ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶)) |
15 | 14 | ssriv 3003 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐵 ∖ 𝐶) ⊆ (𝐵 ∖ ∩ 𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 Vcvv 2601 ∖ cdif 2970 ⊆ wss 2973 ∪ ciun 3678 ∩ ciin 3679 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-in 2979 df-ss 2986 df-iun 3680 df-iin 3681 |
This theorem is referenced by: (None) |
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