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Theorem trin 3855
Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
Assertion
Ref Expression
trin ((Tr A Tr B) → Tr (AB))

Proof of Theorem trin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3120 . . . . 5 (x (AB) ↔ (x A x B))
2 trss 3854 . . . . . 6 (Tr A → (x AxA))
3 trss 3854 . . . . . 6 (Tr B → (x BxB))
42, 3im2anan9 530 . . . . 5 ((Tr A Tr B) → ((x A x B) → (xA xB)))
51, 4syl5bi 141 . . . 4 ((Tr A Tr B) → (x (AB) → (xA xB)))
6 ssin 3153 . . . 4 ((xA xB) ↔ x ⊆ (AB))
75, 6syl6ib 150 . . 3 ((Tr A Tr B) → (x (AB) → x ⊆ (AB)))
87ralrimiv 2385 . 2 ((Tr A Tr B) → x (AB)x ⊆ (AB))
9 dftr3 3849 . 2 (Tr (AB) ↔ x (AB)x ⊆ (AB))
108, 9sylibr 137 1 ((Tr A Tr B) → Tr (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  cin 2910  wss 2911  Tr wtr 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846
This theorem is referenced by:  ordin  4088
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