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Mirrors > Home > MPE Home > Th. List > truni | Structured version Visualization version 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 5146 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 𝑥) | |
2 | uniiun 4941 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
3 | treq 5139 | . . 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 237 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1542 ∀wral 3053 ∪ cuni 4793 ∪ ciun 4878 Tr wtr 5133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3399 df-in 3848 df-ss 3858 df-uni 4794 df-iun 4880 df-tr 5134 |
This theorem is referenced by: dfon2lem1 33323 |
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