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Mirrors > Home > ILE Home > Th. List > triun | GIF version |
Description: The indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
triun | ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliun 3825 | . . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | r19.29 2572 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵)) | |
3 | nfcv 2282 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
4 | nfiu1 3851 | . . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵 | |
5 | 3, 4 | nfss 3095 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 |
6 | trss 4043 | . . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵)) | |
7 | 6 | imp 123 | . . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵) |
8 | ssiun2 3864 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
9 | sstr2 3109 | . . . . . . . 8 ⊢ (𝑦 ⊆ 𝐵 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) | |
10 | 8, 9 | syl5com 29 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ⊆ 𝐵 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
11 | 7, 10 | syl5 32 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)) |
12 | 5, 11 | rexlimi 2545 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
13 | 2, 12 | syl 14 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
14 | 1, 13 | sylan2b 285 | . . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
15 | 14 | ralrimiva 2508 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) |
16 | dftr3 4038 | . 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵) | |
17 | 15, 16 | sylibr 133 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 ∀wral 2417 ∃wrex 2418 ⊆ wss 3076 ∪ ciun 3821 Tr wtr 4034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-in 3082 df-ss 3089 df-uni 3745 df-iun 3823 df-tr 4035 |
This theorem is referenced by: truni 4048 |
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