Theorem triun 4918

Description: The 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)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem triun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4676	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.29 3210	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵))
3		nfcv 2902	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
4		nfiu1 4702	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 𝐵
5	3, 4	nfss 3737	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵
6		trss 4913	. . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵))
7	6	imp 444	. . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵)
8		ssiun2 4715	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
9		sstr2 3751	. . . . . . 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 3162	. . . . 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 493	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
14	13	ralrimiva 3104	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
15		dftr3 4908	. 2 ⊢ (Tr ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵)
16	14, 15	sylibr 224	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∪ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   ⊆ wss 3715  ∪ ciun 4672  Tr wtr 4904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-iun 4674  df-tr 4905

This theorem is referenced by:  truni  4919  r1tr  8814  r1elssi  8843  iunord  42950