Theorem suctr 6440

Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)

Assertion

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

Proof of Theorem suctr

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

Step	Hyp	Ref	Expression
1		elsuci 6421	. . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
2		trel 5238	. . . . . . . 8 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
3	2	expdimp 452	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴))
4		eleq2 2823	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
5	4	biimpcd 249	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴))
6	5	adantl 481	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴))
7	3, 6	jaod 859	. . . . . 6 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴))
8	1, 7	syl5 34	. . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ suc 𝐴 → 𝑧 ∈ 𝐴))
9	8	expimpd 453	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝐴))
10		elelsuc 6427	. . . 4 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)
11	9, 10	syl6 35	. . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
12	11	alrimivv 1928	. 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
13		dftr2 5231	. 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
14	12, 13	sylibr 234	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540   ∈ wcel 2108  Tr wtr 5229  suc csuc 6354

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-ss 3943  df-sn 4602  df-uni 4884  df-tr 5230  df-suc 6358

This theorem is referenced by:  ordsuci  7802  dfon2lem3  35803  dfon2lem7  35807  dford3lem2  43051