Theorem truniALT 45135

Description: The union of a class of transitive sets is transitive. Alternate proof of truni 5235. truniALT 45135 is truniALTVD 45471 without virtual deductions and was automatically derived from truniALTVD 45471 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem truniALT

Dummy variables 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 489	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑦 ∈ ∪ 𝐴)
2	1	a1i 11	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑦 ∈ ∪ 𝐴))
3		eluni 4876	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴))
4	2, 3	imbitrdi 254	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)))
5		simpl 487	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ 𝑦)
6	5	a1i 11	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ 𝑦))
7		simpl 487	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑦 ∈ 𝑞)
8	7	2a1i 12	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑦 ∈ 𝑞)))
9		simpr 489	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)
10	9	2a1i 12	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑞 ∈ 𝐴)))
11		rspsbc 3841	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
12	11	com12 33	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥))
13	10, 12	syl6d 76	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → [𝑞 / 𝑥]Tr 𝑥)))
14		trsbc 45134	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
15	14	biimpd 232	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
16	10, 13, 15	ee33 45115	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → Tr 𝑞)))
17		trel 5227	. . . . . . . . 9 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
18	17	expdcom 419	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → (Tr 𝑞 → 𝑧 ∈ 𝑞)))
19	6, 8, 16, 18	ee233 45113	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ 𝑞)))
20		elunii 4878	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
21	20	ex 417	. . . . . . 7 ⊢ (𝑧 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴))
22	19, 10, 21	ee33 45115	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)))
23	22	alrimdv 1956	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)))
24		19.23v 1969	. . . . 5 ⊢ (∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) ↔ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴))
25	23, 24	imbitrdi 254	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)))
26	4, 25	mpdd 44	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
27	26	alrimivv 1955	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
28		dftr2 5221	. 2 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
29	27, 28	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  [wsbc 3753  ∪ cuni 4873  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3465  df-sbc 3754  df-ss 3930  df-uni 4874  df-tr 5220

This theorem is referenced by: (None)