Theorem suctrALT 45767

Description: The successor of a transitive class is transitive. The proof of https://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 45767 using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/suctrro.html 45767 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 6444 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem suctrALT

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

Step	Hyp	Ref	Expression
1		sssucid 6438	. . . . . . 7 ⊢ 𝐴 ⊆ suc 𝐴
2		id 23	. . . . . . . 8 ⊢ (Tr 𝐴 → Tr 𝐴)
3		id 23	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴))
4	3	simpld 500	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
5		id 23	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐴)
6		trel 5220	. . . . . . . . . 10 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
7	6	3impib 1134	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
8	7	idiALT 45420	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
9	2, 4, 5, 8	syl3an 1178	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
10	1, 9	sselid 3929	. . . . . 6 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ suc 𝐴)
11	10	3expia 1139	. . . . 5 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴))
12	4	adantr 486	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝑦)
13		id 23	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
14	13	adantl 487	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
15	12, 14	eleqtrd 2863	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
16	1, 15	sselid 3929	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
17	16	ex 418	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴))
18	17	adantl 487	. . . . 5 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴))
19	3	simprd 501	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20		elsuci 6425	. . . . . . 7 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
21	19, 20	syl 18	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
22	21	adantl 487	. . . . 5 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
23	11, 18, 22	mpjaod 874	. . . 4 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
24	23	ex 418	. . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
25	24	alrimivv 1961	. 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
26		dftr2 5214	. . 3 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
27	26	biimpri 231	. 2 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
28	25, 27	syl 18	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2145  Tr wtr 5212  suc csuc 6357

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 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453  df-un 3904  df-ss 3916  df-sn 4585  df-uni 4868  df-tr 5213  df-suc 6361

This theorem is used by: (None)