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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 5177 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 𝑥) | |
2 | uniiun 4974 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
3 | treq 5170 | . . 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 236 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∀wral 3138 ∪ cuni 4831 ∪ ciun 4911 Tr wtr 5164 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-v 3496 df-in 3942 df-ss 3951 df-uni 4832 df-iun 4913 df-tr 5165 |
This theorem is referenced by: dfon2lem1 33023 |
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