Theorem suctrALTcfVD 45373

Description: The following User's Proof is a Virtual Deduction proof (see wvd1 45020) 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 45372 is suctrALTcfVD 45373 without virtual deductions and was derived automatically from suctrALTcfVD 45373. 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 6399	. . . . . . . 8 ⊢ 𝐴 ⊆ suc 𝐴
2		idn1 45025	. . . . . . . . 9 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn1 45025	. . . . . . . . . 10 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )
4		simpl 483	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
5	3, 4	el1 45079	. . . . . . . . 9 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ 𝑦   )
6		idn1 45025	. . . . . . . . 9 ⊢ (   𝑦 ∈ 𝐴   ▶   𝑦 ∈ 𝐴   )
7		trel 5194	. . . . . . . . . 10 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
8	7	3impib 1122	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
9	2, 5, 6, 8	el123 45214	. . . . . . . 8 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ 𝐴   )
10		ssel2 3917	. . . . . . . 8 ⊢ ((𝐴 ⊆ suc 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ suc 𝐴)
11	1, 9, 10	el0321old 45167	. . . . . . 7 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
12	11	int3 45063	. . . . . 6 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )   ▶   (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)   )
13		idn1 45025	. . . . . . . . 9 ⊢ (   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
14		eleq2 2829	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
15	14	biimpac 479	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
16	5, 13, 15	el12 45176	. . . . . . . 8 ⊢ (   (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ 𝐴   )
17	1, 16, 10	el021old 45152	. . . . . . 7 ⊢ (   (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
18	17	int2 45057	. . . . . 6 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)   )
19		simpr 485	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20	3, 19	el1 45079	. . . . . . 7 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
21		elsuci 6386	. . . . . . 7 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
22	20, 21	el1 45079	. . . . . 6 ⊢ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)   )
23		jao 968	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24	23	3imp 1116	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
25	12, 18, 22, 24	el2122old 45169	. . . . 5 ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )   ▶   𝑧 ∈ suc 𝐴   )
26	25	int2 45057	. . . 4 ⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
27	26	gen12 45069	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28		dftr2 5188	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
29	28	biimpri 229	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
30	27, 29	el1 45079	. 2 ⊢ (   Tr 𝐴   ▶   Tr suc 𝐴   )
31	30	in1 45022	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853  ∀wal 1545   = wceq 1547   ∈ wcel 2119   ⊆ wss 3890  Tr wtr 5186  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-sn 4563  df-uni 4846  df-tr 5187  df-suc 6323  df-vd1 45021  df-vhc2 45032  df-vhc3 45040

This theorem is referenced by: (None)