Theorem suctrALT2 44857

Description: Virtual deduction proof of suctr 6470. The successor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 44856 using the tools command file translate_without_overwriting_minimize_excluding_duplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
suctrALT2	⊢ (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT2

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

Step	Hyp	Ref	Expression
1		sssucid 6464	. . . . 5 ⊢ 𝐴 ⊆ suc 𝐴
2		trel 5268	. . . . . . 7 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
3	2	expd 415	. . . . . 6 ⊢ (Tr 𝐴 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
4	3	adantrd 491	. . . . 5 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
5		ssel 3977	. . . . 5 ⊢ (𝐴 ⊆ suc 𝐴 → (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴))
6	1, 4, 5	ee03 44761	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)))
7		simpl 482	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
8	7	a1i 11	. . . . . 6 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦))
9		eleq2 2830	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
10	9	biimpcd 249	. . . . . 6 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴))
11	8, 10	syl6 35	. . . . 5 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)))
12	1, 11, 5	ee03 44761	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)))
13		simpr 484	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
14	13	a1i 11	. . . . 5 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴))
15		elsuci 6451	. . . . 5 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
16	14, 15	syl6 35	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
17		jao 963	. . . 4 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
18	6, 12, 16, 17	ee222 44522	. . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
19	18	alrimivv 1928	. 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
20		dftr2 5261	. 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
21	19, 20	sylibr 234	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848  ∀wal 1538   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  Tr wtr 5259  suc csuc 6386

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-uni 4908  df-tr 5260  df-suc 6390

This theorem is referenced by: (None)