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Mirrors > Home > ILE Home > Th. List > truni | GIF version |
Description: The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.) |
Ref | Expression |
---|---|
truni | ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | triun 4075 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 𝑥) | |
2 | uniiun 3902 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
3 | treq 4068 | . . 3 ⊢ (∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 → (Tr ∪ 𝐴 ↔ Tr ∪ 𝑥 ∈ 𝐴 𝑥)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (Tr ∪ 𝐴 ↔ Tr ∪ 𝑥 ∈ 𝐴 𝑥) |
5 | 1, 4 | sylibr 133 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∀wral 2435 ∪ cuni 3772 ∪ ciun 3849 Tr wtr 4062 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-in 3108 df-ss 3115 df-uni 3773 df-iun 3851 df-tr 4063 |
This theorem is referenced by: (None) |
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