Theorem trintALT 45480

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 45480 is an alternate proof of trint 5240. trintALT 45480 is trintALTVD 45479 without virtual deductions and was automatically derived from trintALTVD 45479 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 487	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦)
2	1	a1i 11	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦))
3		iidn3 45101	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑞 ∈ 𝐴)))
4		id 23	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑥 ∈ 𝐴 Tr 𝑥)
5		rspsbc 3841	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
6	3, 4, 5	ee31 45351	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥)))
7		trsbc 45140	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
8	7	biimpd 232	. . . . . . 7 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
9	3, 6, 8	ee33 45121	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → Tr 𝑞)))
10		simpr 489	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴)
11	10	a1i 11	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴))
12		elintg 4924	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
13	12	ibi 270	. . . . . . . 8 ⊢ (𝑦 ∈ ∩ 𝐴 → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞)
14	11, 13	syl6 36	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
15		rsp 3259	. . . . . . 7 ⊢ (∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞))
16	14, 15	syl6 36	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)))
17		trel 5230	. . . . . . 7 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
18	17	expd 420	. . . . . 6 ⊢ (Tr 𝑞 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → 𝑧 ∈ 𝑞)))
19	9, 2, 16, 18	ee323 45108	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)))
20	19	ralrimdv 3169	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
21		elintg 4924	. . . . 5 ⊢ (𝑧 ∈ 𝑦 → (𝑧 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
22	21	biimprd 251	. . . 4 ⊢ (𝑧 ∈ 𝑦 → (∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 → 𝑧 ∈ ∩ 𝐴))
23	2, 20, 22	syl6c 71	. . 3 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
24	23	alrimivv 1955	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
25		dftr2 5224	. 2 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
26	24, 25	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565   ∈ wcel 2149  ∀wral 3085  [wsbc 3753  ∩ cint 4916  Tr wtr 5222

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-3an 1103  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 4877  df-int 4917  df-tr 5223

This theorem is referenced by: (None)