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Theorem truni 4727
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 4726 . 2 (∀𝑥𝐴 Tr 𝑥 → Tr 𝑥𝐴 𝑥)
2 uniiun 4539 . . 3 𝐴 = 𝑥𝐴 𝑥
3 treq 4718 . . 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 1480  wral 2907   cuni 4402   ciun 4485  Tr wtr 4712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403  df-iun 4487  df-tr 4713
This theorem is referenced by:  dfon2lem1  31389
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