Theorem truniALT 44494

Description: The union of a class of transitive sets is transitive. Alternate proof of truni 5255. truniALT 44494 is truniALTVD 44831 without virtual deductions and was automatically derived from truniALTVD 44831 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 484	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑦 ∈ ∪ 𝐴)
2	1	a1i 11	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑦 ∈ ∪ 𝐴))
3		eluni 4890	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴))
4	2, 3	imbitrdi 251	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴)))
5		simpl 482	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ 𝑦)
6	5	a1i 11	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ 𝑦))
7		simpl 482	. . . . . . . . 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 484	. . . . . . . . . 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 3859	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
12	11	com12 32	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥))
13	10, 12	syl6d 75	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → [𝑞 / 𝑥]Tr 𝑥)))
14		trsbc 44493	. . . . . . . . . 10 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
15	14	biimpd 229	. . . . . . . . 9 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
16	10, 13, 15	ee33 44474	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → Tr 𝑞)))
17		trel 5248	. . . . . . . . 9 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
18	17	expdcom 414	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → (Tr 𝑞 → 𝑧 ∈ 𝑞)))
19	6, 8, 16, 18	ee233 44472	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ 𝑞)))
20		elunii 4892	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)
21	20	ex 412	. . . . . . 7 ⊢ (𝑧 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴))
22	19, 10, 21	ee33 44474	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)))
23	22	alrimdv 1928	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → ∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)))
24		19.23v 1941	. . . . 5 ⊢ (∀𝑞((𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴) ↔ (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴))
25	23, 24	imbitrdi 251	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → (∃𝑞(𝑦 ∈ 𝑞 ∧ 𝑞 ∈ 𝐴) → 𝑧 ∈ ∪ 𝐴)))
26	4, 25	mpdd 43	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
27	26	alrimivv 1927	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
28		dftr2 5241	. 2 ⊢ (Tr ∪ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∪ 𝐴) → 𝑧 ∈ ∪ 𝐴))
29	27, 28	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537  ∃wex 1778   ∈ wcel 2107  ∀wral 3050  [wsbc 3770  ∪ cuni 4887  Tr wtr 5239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-v 3465  df-sbc 3771  df-ss 3948  df-uni 4888  df-tr 5240

This theorem is referenced by: (None)