Theorem trintALT 45589

Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 45589 is an alternate proof of trint 5236. trintALT 45589 is trintALTVD 45588 without virtual deductions and was automatically derived from trintALTVD 45588 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 45210	. . . . . . 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 3832	. . . . . . . 8 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
6	3, 4, 5	ee31 45460	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → [𝑞 / 𝑥]Tr 𝑥)))
7		trsbc 45249	. . . . . . . 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 45230	. . . . . 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 4920	. . . . . . . . 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 3253	. . . . . . 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 5226	. . . . . . 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 45217	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)))
20	19	ralrimdv 3163	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
21		elintg 4920	. . . . 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 1958	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
25		dftr2 5220	. 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 1568   ∈ wcel 2143  ∀wral 3079  [wsbc 3744  ∩ cint 4912  Tr wtr 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-sbc 3745  df-ss 3922  df-uni 4873  df-int 4913  df-tr 5219

This theorem is referenced by: (None)