Theorem triun 5227

Description: An indexed union of a class of transitive sets is transitive. (Contributed by Mario Carneiro, 16-Nov-2014.)

Assertion

Ref	Expression
triun	⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem triun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4956	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.29 3128	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵))
3		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
4		nfiu1 4988	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
5	3, 4	nfss 3932	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
6		trss 5222	. . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵))
7	6	imp 411	. . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵)
8		ssiun2 5008	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9		sstr2 3946	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐵 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
10	7, 8, 9	syl2imc 42	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
11	5, 10	rexlimi 3265	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
12	2, 11	syl 18	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1, 12	sylan2b 605	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
14	13	ralrimiva 3157	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
15		dftr3 5217	. 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
16	14, 15	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907  ∪ ciun 4952  Tr wtr 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-v 3459  df-ss 3924  df-uni 4869  df-iun 4954  df-tr 5213

This theorem is referenced by:  truni  5228  r1tr  9736  r1elssi  9765  tz9.1regs  35442  ttciunun  36884  iunord  50305