Theorem triun 5229

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 4959	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.29 3094	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵))
3		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
4		nfiu1 4991	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
5	3, 4	nfss 3939	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
6		trss 5225	. . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵))
7	6	imp 406	. . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵)
8		ssiun2 5011	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9		sstr2 3953	. . . . . . 7 ⊢ (𝑦 ⊆ 𝐵 → (𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
10	7, 8, 9	syl2imc 41	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵))
11	5, 10	rexlimi 3237	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
12	2, 11	syl 17	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1, 12	sylan2b 594	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
14	13	ralrimiva 3125	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
15		dftr3 5220	. 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
16	14, 15	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∪ ciun 4955  Tr wtr 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3449  df-ss 3931  df-uni 4872  df-iun 4957  df-tr 5215

This theorem is referenced by:  truni  5230  r1tr  9729  r1elssi  9758  iunord  49665