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Theorem truni 4901
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 4900 . 2 (∀𝑥𝐴 Tr 𝑥 → Tr 𝑥𝐴 𝑥)
2 uniiun 4708 . . 3 𝐴 = 𝑥𝐴 𝑥
3 treq 4893 . . 3 ( 𝐴 = 𝑥𝐴 𝑥 → (Tr 𝐴 ↔ Tr 𝑥𝐴 𝑥))
42, 3ax-mp 5 . 2 (Tr 𝐴 ↔ Tr 𝑥𝐴 𝑥)
51, 4sylibr 224 1 (∀𝑥𝐴 Tr 𝑥 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wral 3061   cuni 4575   ciun 4655  Tr wtr 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-uni 4576  df-iun 4657  df-tr 4888
This theorem is referenced by:  dfon2lem1  32024
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