Theorem trintALTVD 45703

Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 45704. 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 45704 is trintALTVD 45703 without virtual deductions and was automatically derived from trintALTVD 45703.

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 45437	. . . . . . 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 45455	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4		idn3 45439	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   )
5		idn1 45398	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
6		rspsbc 3826	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
7	4, 5, 6	e31 45574	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
8		trsbc 45364	. . . . . . . . . . . 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 45557	. . . . . . . . . 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 45455	. . . . . . . . . . . . 13 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑦 ∈ ∩ 𝐴   )
13		elintg 4915	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ∩ 𝐴 → (𝑦 ∈ ∩ 𝐴 ↔ ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞))
14	13	ibi 270	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∩ 𝐴 → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞)
15	12, 14	e2 45455	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞   )
16		rsp 3250	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞))
17	15, 16	e2 45455	. . . . . . . . . . 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 45581	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   )
20		trel 5220	. . . . . . . . . . 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 45589	. . . . . . . . 9 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   )
23	22	in3 45433	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
24	23	gen21 45443	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
25		df-ral 3077	. . . . . . . 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 45455	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞   )
28		elintg 4915	. . . . . . 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 45495	. . . . 5 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   )
31	30	in2 45429	. . . 4 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
32	31	gen12 45442	. . 3 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
33		dftr2 5214	. . . 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 45451	. 2 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   Tr ∩ 𝐴   )
36	35	in1 45395	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   ∈ wcel 2145  ∀wral 3076  [wsbc 3739  ∩ cint 4907  Tr wtr 5212

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 2213  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-sbc 3740  df-ss 3916  df-uni 4868  df-int 4908  df-tr 5213  df-vd1 45394  df-vd2 45402  df-vd3 45414

This theorem is used by: (None)