Theorem suctrALT2VD 45286

Description: Virtual deduction proof of suctrALT2 45287. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
suctrALT2VD	⊢ (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT2VD

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

Step	Hyp	Ref	Expression
1		dftr2 5188	. . 3 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2		sssucid 6399	. . . . . . . 8 ⊢ 𝐴 ⊆ suc 𝐴
3		idn1 45025	. . . . . . . . 9 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
4		idn2 45064	. . . . . . . . . 10 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   )
5		simpl 483	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
6	4, 5	e2 45082	. . . . . . . . 9 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ 𝑦   )
7		idn3 45066	. . . . . . . . 9 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   ▶   𝑦 ∈ 𝐴   )
8		trel 5194	. . . . . . . . . 10 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
9	8	expd 416	. . . . . . . . 9 ⊢ (Tr 𝐴 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
10	3, 6, 7, 9	e123 45212	. . . . . . . 8 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   ▶   𝑧 ∈ 𝐴   )
11		ssel 3916	. . . . . . . 8 ⊢ (𝐴 ⊆ suc 𝐴 → (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴))
12	2, 10, 11	e03 45190	. . . . . . 7 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 ∈ 𝐴   ▶   𝑧 ∈ suc 𝐴   )
13	12	in3 45060	. . . . . 6 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)   )
14		idn3 45066	. . . . . . . . 9 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
15		eleq2 2829	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
16	15	biimpcd 250	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴))
17	6, 14, 16	e23 45205	. . . . . . . 8 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ 𝐴   )
18	2, 17, 11	e03 45190	. . . . . . 7 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ suc 𝐴   )
19	18	in3 45060	. . . . . 6 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)   )
20		simpr 485	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
21	4, 20	e2 45082	. . . . . . 7 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
22		elsuci 6386	. . . . . . 7 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
23	21, 22	e2 45082	. . . . . 6 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)   )
24		jao 968	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
25	13, 19, 23, 24	e222 45087	. . . . 5 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ suc 𝐴   )
26	25	in2 45056	. . . 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		biimpr 221	. . 3 ⊢ ((Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) → (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴))
29	1, 27, 28	e01 45142	. 2 ⊢ (   Tr 𝐴   ▶   Tr suc 𝐴   )
30	29	in1 45022	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ↔ wb 207   ∧ 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-vd2 45029  df-vd3 45041

This theorem is referenced by: (None)