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Mirrors > Home > MPE Home > Th. List > trint | Structured version Visualization version GIF version |
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. (Contributed by Scott Fenton, 25-Feb-2011.) (Proof shortened by BJ, 3-Oct-2022.) |
Ref | Expression |
---|---|
trint | ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | triin 4990 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 4794 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | treq 4981 | . . 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 226 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∀wral 3117 ∩ cint 4697 ∩ ciin 4741 Tr wtr 4975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-v 3416 df-in 3805 df-ss 3812 df-uni 4659 df-int 4698 df-iin 4743 df-tr 4976 |
This theorem is referenced by: tctr 8893 intwun 9872 intgru 9951 dfon2lem8 32222 |
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