![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5286 | . 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝑥 ∈ 𝐴 𝑥) | |
2 | intiin 5066 | . . 3 ⊢ ∩ 𝐴 = ∩ 𝑥 ∈ 𝐴 𝑥 | |
3 | treq 5277 | . . 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 233 | 1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∀wral 3050 ∩ cint 4953 ∩ ciin 5001 Tr wtr 5269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-v 3463 df-ss 3963 df-uni 4913 df-int 4954 df-iin 5003 df-tr 5270 |
This theorem is referenced by: tctr 9779 intwun 10774 intgru 10853 dfon2lem8 35574 |
Copyright terms: Public domain | W3C validator |