Theorem truni 5235

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	⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem truni

Step	Hyp	Ref	Expression
1		triun 5234	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 𝑥)
2		uniiun 5024	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3		treq 5226	. . 3 ⊢ (∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 → (Tr ∪ 𝐴 ↔ Tr ∪ 𝑥 ∈ 𝐴 𝑥))
4	2, 3	ax-mp 5	. 2 ⊢ (Tr ∪ 𝐴 ↔ Tr ∪ 𝑥 ∈ 𝐴 𝑥)
5	1, 4	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  ∀wral 3079  ∪ cuni 4873  ∪ ciun 4957  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3457  df-ss 3923  df-uni 4874  df-iun 4959  df-tr 5220

This theorem is referenced by:  dfon2lem1  36254