Theorem trintALTVD 44869

Description: The intersection of a class of transitive sets is transitive. Virtual deduction proof of trintALT 44870. 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 44870 is trintALTVD 44869 without virtual deductions and was automatically derived from trintALTVD 44869.

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 44603	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   )
2		simpl 482	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ 𝑦)
3	1, 2	e2 44621	. . . . . 6 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ 𝑦   )
4		idn3 44605	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑞 ∈ 𝐴   )
5		idn1 44564	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑥 ∈ 𝐴 Tr 𝑥   )
6		rspsbc 3842	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 Tr 𝑥 → [𝑞 / 𝑥]Tr 𝑥))
7	4, 5, 6	e31 44740	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   [𝑞 / 𝑥]Tr 𝑥   )
8		trsbc 44530	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 ↔ Tr 𝑞))
9	8	biimpd 229	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝐴 → ([𝑞 / 𝑥]Tr 𝑥 → Tr 𝑞))
10	4, 7, 9	e33 44723	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   Tr 𝑞   )
11		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑦 ∈ ∩ 𝐴)
12	1, 11	e2 44621	. . . . . . . . . . . . 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 267	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ∩ 𝐴 → ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞)
15	12, 14	e2 44621	. . . . . . . . . . . 12 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞   )
16		rsp 3225	. . . . . . . . . . . 12 ⊢ (∀𝑞 ∈ 𝐴 𝑦 ∈ 𝑞 → (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞))
17	15, 16	e2 44621	. . . . . . . . . . 11 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞)   )
18		pm2.27 42	. . . . . . . . . . 11 ⊢ (𝑞 ∈ 𝐴 → ((𝑞 ∈ 𝐴 → 𝑦 ∈ 𝑞) → 𝑦 ∈ 𝑞))
19	4, 17, 18	e32 44747	. . . . . . . . . 10 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑦 ∈ 𝑞   )
20		trel 5223	. . . . . . . . . . 11 ⊢ (Tr 𝑞 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑞) → 𝑧 ∈ 𝑞))
21	20	expd 415	. . . . . . . . . 10 ⊢ (Tr 𝑞 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝑞 → 𝑧 ∈ 𝑞)))
22	10, 3, 19, 21	e323 44755	. . . . . . . . 9 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ,   𝑞 ∈ 𝐴   ▶   𝑧 ∈ 𝑞   )
23	22	in3 44599	. . . . . . . 8 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   (𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
24	23	gen21 44609	. . . . . . 7 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞)   )
25		df-ral 3045	. . . . . . . 8 ⊢ (∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 ↔ ∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞))
26	25	biimpri 228	. . . . . . 7 ⊢ (∀𝑞(𝑞 ∈ 𝐴 → 𝑧 ∈ 𝑞) → ∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞)
27	24, 26	e2 44621	. . . . . 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 248	. . . . . 6 ⊢ (𝑧 ∈ 𝑦 → (∀𝑞 ∈ 𝐴 𝑧 ∈ 𝑞 → 𝑧 ∈ ∩ 𝐴))
30	3, 27, 29	e22 44661	. . . . 5 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴)   ▶   𝑧 ∈ ∩ 𝐴   )
31	30	in2 44595	. . . 4 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
32	31	gen12 44608	. . 3 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴)   )
33		dftr2 5216	. . . 4 ⊢ (Tr ∩ 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴))
34	33	biimpri 228	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ ∩ 𝐴) → 𝑧 ∈ ∩ 𝐴) → Tr ∩ 𝐴)
35	32, 34	e1a 44617	. 2 ⊢ (   ∀𝑥 ∈ 𝐴 Tr 𝑥   ▶   Tr ∩ 𝐴   )
36	35	in1 44561	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝑥 → Tr ∩ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  ∀wral 3044  [wsbc 3753  ∩ cint 4910  Tr wtr 5214

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3449  df-sbc 3754  df-ss 3931  df-uni 4872  df-int 4911  df-tr 5215  df-vd1 44560  df-vd2 44568  df-vd3 44580

This theorem is referenced by: (None)