Theorem trintALT 42501

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 42501 is an alternate proof of trint 5207. trintALT 42501 is trintALTVD 42500 without virtual deductions and was automatically derived from trintALTVD 42500 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZE_WITH *" command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
trintALT	⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem trintALT

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

Step	Hyp	Ref	Expression
1		simpl 483	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦)
2	1	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦))
3		iidn3 42121	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑞 ∈ 𝐴)))
4		id 22	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑥 ∈ 𝐴 Tr 𝑥)
5		rspsbc 3812	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
6	3, 4, 5	ee31 42372	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥)))
7		trsbc 42160	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
8	7	biimpd 228	. . . . . . 7 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
9	3, 6, 8	ee33 42141	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → Tr 𝑞)))
10		simpr 485	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴)
11	10	a1i 11	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴))
12		elintg 4887	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
13	12	ibi 266	. . . . . . . 8 ⊢ (𝑦 ∈ ∩ 𝐴 → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞)
14	11, 13	syl6 35	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
15		rsp 3131	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞))
16	14, 15	syl6 35	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)))
17		trel 5198	. . . . . . 7 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
18	17	expd 416	. . . . . 6 ⊢ (Tr 𝑞 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → 𝑧 ∈ 𝑞)))
19	9, 2, 16, 18	ee323 42128	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)))
20	19	ralrimdv 3105	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
21		elintg 4887	. . . . 5 ⊢ (𝑧 ∈ 𝑦 → (𝑧 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
22	21	biimprd 247	. . . 4 ⊢ (𝑧 ∈ 𝑦 → (∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 → 𝑧 ∈ ∩ 𝐴))
23	2, 20, 22	syl6c 70	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
24	23	alrimivv 1931	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
25		dftr2 5193	. 2 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
26	24, 25	sylibr 233	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   ∈ wcel 2106  ∀wral 3064  [wsbc 3716  ∩ cint 4879  Tr wtr 5191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-v 3434  df-sbc 3717  df-in 3894  df-ss 3904  df-uni 4840  df-int 4880  df-tr 5192

This theorem is referenced by: (None)