Theorem trintALT 44905

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 44905 is an alternate proof of trint 5247. trintALT 44905 is trintALTVD 44904 without virtual deductions and was automatically derived from trintALTVD 44904 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 482	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦)
2	1	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦))
3		iidn3 44526	. . . . . . 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 3854	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
6	3, 4, 5	ee31 44776	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥)))
7		trsbc 44565	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
8	7	biimpd 229	. . . . . . 7 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
9	3, 6, 8	ee33 44546	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → Tr 𝑞)))
10		simpr 484	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴)
11	10	a1i 11	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴))
12		elintg 4930	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
13	12	ibi 267	. . . . . . . 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 3230	. . . . . . 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 5238	. . . . . . 7 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
18	17	expd 415	. . . . . 6 ⊢ (Tr 𝑞 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → 𝑧 ∈ 𝑞)))
19	9, 2, 16, 18	ee323 44533	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)))
20	19	ralrimdv 3138	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
21		elintg 4930	. . . . 5 ⊢ (𝑧 ∈ 𝑦 → (𝑧 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
22	21	biimprd 248	. . . 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 1928	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
25		dftr2 5231	. 2 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
26	24, 25	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2108  ∀wral 3051  [wsbc 3765  ∩ cint 4922  Tr wtr 5229

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-v 3461  df-sbc 3766  df-ss 3943  df-uni 4884  df-int 4923  df-tr 5230

This theorem is referenced by: (None)