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Theorem trint 4047
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 (∀𝑥𝐴 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 4036 . . . . . 6 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
21ralbii 2444 . . . . 5 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
32biimpi 119 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
4 df-ral 2422 . . . . . 6 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦𝑥))
54ralbii 2444 . . . . 5 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥))
6 ralcom4 2711 . . . . 5 (∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
75, 6bitri 183 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
83, 7sylib 121 . . 3 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
9 ralim 2494 . . . 4 (∀𝑥𝐴 (𝑦𝑥𝑦𝑥) → (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
109alimi 1432 . . 3 (∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥) → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
118, 10syl 14 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
12 dftr3 4036 . . 3 (Tr 𝐴 ↔ ∀𝑦 𝐴𝑦 𝐴)
13 df-ral 2422 . . . 4 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(𝑦 𝐴𝑦 𝐴))
14 vex 2692 . . . . . . 7 𝑦 ∈ V
1514elint2 3784 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
16 ssint 3793 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
1715, 16imbi12i 238 . . . . 5 ((𝑦 𝐴𝑦 𝐴) ↔ (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1817albii 1447 . . . 4 (∀𝑦(𝑦 𝐴𝑦 𝐴) ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1913, 18bitri 183 . . 3 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
2012, 19bitri 183 . 2 (Tr 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
2111, 20sylibr 133 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1330  wcel 1481  wral 2417  wss 3074   cint 3777  Tr wtr 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3080  df-ss 3087  df-uni 3743  df-int 3778  df-tr 4033
This theorem is referenced by:  onintonm  4439
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