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Theorem trint 3896
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 3885 . . . . . 6 (Tr 𝑥 ↔ ∀𝑦𝑥 𝑦𝑥)
21ralbii 2347 . . . . 5 (∀𝑥𝐴 Tr 𝑥 ↔ ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
32biimpi 117 . . . 4 (∀𝑥𝐴 Tr 𝑥 → ∀𝑥𝐴𝑦𝑥 𝑦𝑥)
4 df-ral 2328 . . . . . 6 (∀𝑦𝑥 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦𝑥))
54ralbii 2347 . . . . 5 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥))
6 ralcom4 2593 . . . . 5 (∀𝑥𝐴𝑦(𝑦𝑥𝑦𝑥) ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
75, 6bitri 177 . . . 4 (∀𝑥𝐴𝑦𝑥 𝑦𝑥 ↔ ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
83, 7sylib 131 . . 3 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥))
9 ralim 2397 . . . 4 (∀𝑥𝐴 (𝑦𝑥𝑦𝑥) → (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
109alimi 1360 . . 3 (∀𝑦𝑥𝐴 (𝑦𝑥𝑦𝑥) → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
118, 10syl 14 . 2 (∀𝑥𝐴 Tr 𝑥 → ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
12 dftr3 3885 . . 3 (Tr 𝐴 ↔ ∀𝑦 𝐴𝑦 𝐴)
13 df-ral 2328 . . . 4 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(𝑦 𝐴𝑦 𝐴))
14 vex 2577 . . . . . . 7 𝑦 ∈ V
1514elint2 3649 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
16 ssint 3658 . . . . . 6 (𝑦 𝐴 ↔ ∀𝑥𝐴 𝑦𝑥)
1715, 16imbi12i 232 . . . . 5 ((𝑦 𝐴𝑦 𝐴) ↔ (∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1817albii 1375 . . . 4 (∀𝑦(𝑦 𝐴𝑦 𝐴) ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
1913, 18bitri 177 . . 3 (∀𝑦 𝐴𝑦 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
2012, 19bitri 177 . 2 (Tr 𝐴 ↔ ∀𝑦(∀𝑥𝐴 𝑦𝑥 → ∀𝑥𝐴 𝑦𝑥))
2111, 20sylibr 141 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wcel 1409  wral 2323  wss 2944   cint 3642  Tr wtr 3881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2951  df-ss 2958  df-uni 3608  df-int 3643  df-tr 3882
This theorem is referenced by:  onintonm  4270
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