Theorem trintALTVD 45710

Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 45711. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. trintALT 45711 is trintALTVD 45710 without virtual deductions and was automatically derived from trintALTVD 45710.

1::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴Tr 𝑥   )
2::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   )
3:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4:2:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑦 ∈ ∩ 𝐴   )
5:4:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑦 ∈ 𝑞   )
6:5:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)   )
7::	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   )
8:7,6:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   )
9:7,1:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
10:7,9:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   Tr 𝑞   )
11:10,3,8:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴), 𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   )
12:11:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
13:12:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
14:13:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴𝑧 ∈ 𝑞   )
15:3,14:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   )
16:15:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
17:16:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
18:17:	⊢ (   ∀𝑥 ∈ 𝐴Tr 𝑥   ▶   Tr ∩ 𝐴   )
qed:18:	⊢ (∀𝑥 ∈ 𝐴Tr 𝑥 → Tr ∩ 𝐴)

(Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem trintALTVD

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

Step	Hyp	Ref	Expression
1		idn2 45444	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   )
2		simpl 488	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦)
3	1, 2	e2 45462	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4		idn3 45446	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   )
5		idn1 45405	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
6		rspsbc 3829	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
7	4, 5, 6	e31 45581	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
8		trsbc 45371	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
9	8	biimpd 232	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
10	4, 7, 9	e33 45564	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   Tr 𝑞   )
11		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴)
12	1, 11	e2 45462	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑦 ∈ ∩ 𝐴   )
13		elintg 4918	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
14	13	ibi 270	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∩ 𝐴 → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞)
15	12, 14	e2 45462	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞   )
16		rsp 3252	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞))
17	15, 16	e2 45462	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)   )
18		pm2.27 43	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝐴 → ((𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞) → 𝑦 ∈ 𝑞))
19	4, 17, 18	e32 45588	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   )
20		trel 5224	. . . . . . . . . . 11 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
21	20	expd 421	. . . . . . . . . 10 ⊢ (Tr 𝑞 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → 𝑧 ∈ 𝑞)))
22	10, 3, 19, 21	e323 45596	. . . . . . . . 9 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   )
23	22	in3 45440	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
24	23	gen21 45450	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
25		df-ral 3079	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 ↔ ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞))
26	25	biimpri 231	. . . . . . 7 ⊢ (∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) → ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞)
27	24, 26	e2 45462	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞   )
28		elintg 4918	. . . . . . 7 ⊢ (𝑧 ∈ 𝑦 → (𝑧 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞))
29	28	biimprd 251	. . . . . 6 ⊢ (𝑧 ∈ 𝑦 → (∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 → 𝑧 ∈ ∩ 𝐴))
30	3, 27, 29	e22 45502	. . . . 5 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   )
31	30	in2 45436	. . . 4 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
32	31	gen12 45449	. . 3 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
33		dftr2 5218	. . . 4 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
34	33	biimpri 231	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) → Tr ∩ 𝐴)
35	32, 34	e1a 45458	. 2 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   Tr ∩ 𝐴   )
36	35	in1 45402	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∀wral 3078  [wsbc 3742  ∩ cint 4910  Tr wtr 5216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-sbc 3743  df-ss 3919  df-uni 4871  df-int 4911  df-tr 5217  df-vd1 45401  df-vd2 45409  df-vd3 45421

This theorem is used by: (None)