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Theorem truni 4076
Description: The union of a class of transitive sets is transitive. Exercise 5(a) of [Enderton] p. 73. (Contributed by Scott Fenton, 21-Feb-2011.) (Proof shortened by Mario Carneiro, 26-Apr-2014.)
Assertion
Ref Expression
truni (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem truni
StepHypRef Expression
1 triun 4075 . 2 (∀𝑥𝐴 Tr 𝑥 → Tr 𝑥𝐴 𝑥)
2 uniiun 3902 . . 3 𝐴 = 𝑥𝐴 𝑥
3 treq 4068 . . 3 ( 𝐴 = 𝑥𝐴 𝑥 → (Tr 𝐴 ↔ Tr 𝑥𝐴 𝑥))
42, 3ax-mp 5 . 2 (Tr 𝐴 ↔ Tr 𝑥𝐴 𝑥)
51, 4sylibr 133 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1335  wral 2435   cuni 3772   ciun 3849  Tr wtr 4062
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-iun 3851  df-tr 4063
This theorem is referenced by: (None)
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