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Theorem trint 3860
 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.)
Assertion
Ref Expression
trint (x A Tr x → Tr A)
Distinct variable group:   x,A

Proof of Theorem trint
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dftr3 3849 . . . . . 6 (Tr xy x yx)
21ralbii 2324 . . . . 5 (x A Tr xx A y x yx)
32biimpi 113 . . . 4 (x A Tr xx A y x yx)
4 df-ral 2305 . . . . . 6 (y x yxy(y xyx))
54ralbii 2324 . . . . 5 (x A y x yxx A y(y xyx))
6 ralcom4 2570 . . . . 5 (x A y(y xyx) ↔ yx A (y xyx))
75, 6bitri 173 . . . 4 (x A y x yxyx A (y xyx))
83, 7sylib 127 . . 3 (x A Tr xyx A (y xyx))
9 ralim 2374 . . . 4 (x A (y xyx) → (x A y xx A yx))
109alimi 1341 . . 3 (yx A (y xyx) → y(x A y xx A yx))
118, 10syl 14 . 2 (x A Tr xy(x A y xx A yx))
12 dftr3 3849 . . 3 (Tr Ay Ay A)
13 df-ral 2305 . . . 4 (y Ay Ay(y Ay A))
14 vex 2554 . . . . . . 7 y V
1514elint2 3613 . . . . . 6 (y Ax A y x)
16 ssint 3622 . . . . . 6 (y Ax A yx)
1715, 16imbi12i 228 . . . . 5 ((y Ay A) ↔ (x A y xx A yx))
1817albii 1356 . . . 4 (y(y Ay A) ↔ y(x A y xx A yx))
1913, 18bitri 173 . . 3 (y Ay Ay(x A y xx A yx))
2012, 19bitri 173 . 2 (Tr Ay(x A y xx A yx))
2111, 20sylibr 137 1 (x A Tr x → Tr A)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1240   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911  ∩ cint 3606  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-bndl 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-int 3607  df-tr 3846 This theorem is referenced by: (None)
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