Theorem suctrALT2 45667

Description: Virtual deduction proof of suctr 6450. The successor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 45666 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 6444	. . . . 5 ⊢ 𝐴 ⊆ suc 𝐴
2		trel 5224	. . . . . . 7 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
3	2	expd 421	. . . . . 6 ⊢ (Tr 𝐴 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
4	3	adantrd 497	. . . . 5 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
5		ssel 3928	. . . . 5 ⊢ (𝐴 ⊆ suc 𝐴 → (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴))
6	1, 4, 5	ee03 45571	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)))
7		simpl 488	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
8	7	a1i 11	. . . . . 6 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦))
9		eleq2 2851	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
10	9	biimpcd 252	. . . . . 6 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴))
11	8, 10	syl6 36	. . . . 5 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)))
12	1, 11, 5	ee03 45571	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)))
13		simpr 490	. . . . . 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 6431	. . . . 5 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
16	14, 15	syl6 36	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)))
17		jao 975	. . . 4 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
18	6, 12, 16, 17	ee222 45333	. . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
19	18	alrimivv 1961	. 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
20		dftr2 5218	. 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
21	19, 20	sylibr 237	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902  Tr wtr 5216  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-uni 4871  df-tr 5217  df-suc 6367

This theorem is used by: (None)