Theorem truni 5205

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 5204	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝑥 ∈ 𝐴 𝑥)
2		uniiun 4988	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3		treq 5197	. . 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 233	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ∀wral 3064  ∪ cuni 4839  ∪ ciun 4924  Tr wtr 5191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-iun 4926  df-tr 5192

This theorem is referenced by:  dfon2lem1  33759