Theorem triun 5274

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 4995	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.29 3114	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵))
3		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
4		nfiu1 5027	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
5	3, 4	nfss 3976	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
6		trss 5270	. . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵))
7	6	imp 406	. . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵)
8		ssiun2 5047	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9		sstr2 3990	. . . . . . 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 3259	. . . . 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 3146	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
15		dftr3 5265	. 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 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ∪ ciun 4991  Tr wtr 5259

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-iun 4993  df-tr 5260

This theorem is referenced by:  truni  5275  r1tr  9816  r1elssi  9845  iunord  49195