Theorem suctr 4399

Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)

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		simpr 109	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
2		vex 2729	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	elsuc 4384	. . . . 5 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
4	1, 3	sylib 121	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
5		simpl 108	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
6		eleq2 2230	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
7	5, 6	syl5ibcom 154	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴))
8		elelsuc 4387	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)
9	7, 8	syl6 33	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴))
10		trel 4087	. . . . . . . . 9 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
11	10	expd 256	. . . . . . . 8 ⊢ (Tr 𝐴 → (𝑧 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
12	11	adantrd 277	. . . . . . 7 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)))
13	12, 8	syl8 71	. . . . . 6 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)))
14		jao 745	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
15	13, 14	syl6 33	. . . . 5 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴))))
16	9, 15	mpdi 43	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
17	4, 16	mpdi 43	. . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
18	17	alrimivv 1863	. 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
19		dftr2 4082	. 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
20	18, 19	sylibr 133	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  ∀wal 1341   = wceq 1343   ∈ wcel 2136  Tr wtr 4080  suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-uni 3790  df-tr 4081  df-suc 4349

This theorem is referenced by:  ordsucim  4477  ordom  4584