Theorem suctrALTcfVD 44954

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 44601) using conjunction-form virtual hypothesis collections. The conjunction-form version of completeusersproof.cmd. It allows the User to avoid superflous virtual hypotheses. This proof was completed automatically by a tools program which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 44953 is suctrALTcfVD 44954 without virtual deductions and was derived automatically from suctrALTcfVD 44954. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

1::	⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
2::	⊢ (   ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )
3:2:	⊢ (   ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ 𝑦   )
4::	⊢ (   ................................... ....... 𝑦 ∈ 𝐴   ▶   𝑦 ∈ 𝐴   )
5:1,3,4:	⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) , 𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ 𝐴   )
6::	⊢ 𝐴 ⊆ suc 𝐴
7:5,6:	⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) , 𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
8:7:	⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)    )   ▶   (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)   )
9::	⊢ (   ................................... ...... 𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
10:3,9:	⊢ (   ........ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ 𝐴   )
11:10,6:	⊢ (   ........ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
12:11:	⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)   )
13:2:	⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
14:13:	⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)   )
15:8,12,14:	⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)    )   ▶   𝑧 ∈ suc 𝐴   )
16:15:	⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
17:16:	⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
18:17:	⊢ (   Tr 𝐴   ▶   Tr suc 𝐴   )
qed:18:	⊢ (Tr 𝐴 → Tr suc 𝐴)

Assertion

Ref	Expression
suctrALTcfVD	⊢ (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALTcfVD

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

Step	Hyp	Ref	Expression
1		sssucid 6388	. . . . . . . 8 ⊢ 𝐴 ⊆ suc 𝐴
2		idn1 44606	. . . . . . . . 9 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn1 44606	. . . . . . . . . 10 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )
4		simpl 482	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
5	3, 4	el1 44660	. . . . . . . . 9 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ 𝑦   )
6		idn1 44606	. . . . . . . . 9 ⊢ (   𝑦 ∈ 𝐴   ▶   𝑦 ∈ 𝐴   )
7		trel 5206	. . . . . . . . . 10 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
8	7	3impib 1116	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
9	2, 5, 6, 8	el123 44795	. . . . . . . 8 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ 𝐴   )
10		ssel2 3929	. . . . . . . 8 ⊢ ((𝐴 ⊆ suc 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ suc 𝐴)
11	1, 9, 10	el0321old 44748	. . . . . . 7 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
12	11	int3 44644	. . . . . 6 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )   ▶   (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)   )
13		idn1 44606	. . . . . . . . 9 ⊢ (   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
14		eleq2 2820	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
15	14	biimpac 478	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
16	5, 13, 15	el12 44757	. . . . . . . 8 ⊢ (   (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ 𝐴   )
17	1, 16, 10	el021old 44733	. . . . . . 7 ⊢ (   (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
18	17	int2 44638	. . . . . 6 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)   )
19		simpr 484	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20	3, 19	el1 44660	. . . . . . 7 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
21		elsuci 6375	. . . . . . 7 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
22	20, 21	el1 44660	. . . . . 6 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)   )
23		jao 962	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24	23	3imp 1110	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
25	12, 18, 22, 24	el2122old 44750	. . . . 5 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )   ▶   𝑧 ∈ suc 𝐴   )
26	25	int2 44638	. . . 4 ⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
27	26	gen12 44650	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28		dftr2 5200	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
29	28	biimpri 228	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
30	27, 29	el1 44660	. 2 ⊢ (   Tr 𝐴   ▶   Tr suc 𝐴   )
31	30	in1 44603	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2111   ⊆ wss 3902  Tr wtr 5198  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3907  df-ss 3919  df-sn 4577  df-uni 4860  df-tr 5199  df-suc 6312  df-vd1 44602  df-vhc2 44613  df-vhc3 44621

This theorem is referenced by: (None)