Theorem suctrALT3 44600

Description: The successor of a transitive class is transitive. suctrALT3 44600 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/suctralt3vd.html 44600. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 44245 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 19 used jaoded 44242). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 24 used dftr2 5272) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
suctrALT3	⊢ (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT3

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

Step	Hyp	Ref	Expression
1		sssucid 6456	. . . . . . . . 9 ⊢ 𝐴 ⊆ suc 𝐴
2		id 22	. . . . . . . . . 10 ⊢ (Tr 𝐴 → Tr 𝐴)
3		id 22	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴))
4	3	simpld 493	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
5		id 22	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐴)
6	2, 4, 5	trelded 44241	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
7	1, 6	sselid 3977	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ suc 𝐴)
8	7	3expia 1118	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴))
9		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
10		eleq2 2815	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
11	10	biimpac 477	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
12	4, 9, 11	syl2an 594	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
13	1, 12	sselid 3977	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
14	13	ex 411	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴))
15	3	simprd 494	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
16		elsuci 6443	. . . . . . . 8 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
18	8, 14, 17	jaoded 44242	. . . . . 6 ⊢ (((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
19	18	un2122 44466	. . . . 5 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
20	19	ex 411	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
21	20	alrimivv 1924	. . 3 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
22		dftr2 5272	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
23	22	biimpri 227	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
24	21, 23	syl 17	. 2 ⊢ (Tr 𝐴 → Tr suc 𝐴)
25	24	idiALT 44153	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   ∧ w3a 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099  Tr wtr 5270  suc csuc 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-un 3952  df-ss 3964  df-sn 4634  df-uni 4914  df-tr 5271  df-suc 6382

This theorem is referenced by: (None)