Theorem suctrALTcf 41246

Description: The sucessor of a transitive class is transitive. suctrALTcf 41246, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 41247, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem suctrALTcf

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

Step	Hyp	Ref	Expression
1		sssucid 6261	. . . . . . . 8 ⊢ 𝐴 ⊆ suc 𝐴
2		id 22	. . . . . . . . 9 ⊢ (Tr 𝐴 → Tr 𝐴)
3		id 22	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴))
4		simpl 485	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
5	3, 4	syl 17	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝑦)
6		id 22	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐴)
7		trel 5170	. . . . . . . . . 10 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
8	7	3impib 1110	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
9	2, 5, 6, 8	syl3an 1154	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)
10		ssel2 3960	. . . . . . . 8 ⊢ ((𝐴 ⊆ suc 𝐴 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ suc 𝐴)
11	1, 9, 10	eel0321old 41040	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ suc 𝐴)
12	11	3expia 1115	. . . . . 6 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴))
13		id 22	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
14		eleq2 2899	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴))
15	14	biimpac 481	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
16	5, 13, 15	syl2an 597	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)
17	1, 16, 10	eel021old 41024	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
18	17	ex 415	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴))
19		simpr 487	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20	3, 19	syl 17	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
21		elsuci 6250	. . . . . . 7 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
22	20, 21	syl 17	. . . . . 6 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
23		jao 956	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24	23	3imp 1105	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) ∧ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
25	12, 18, 22, 24	eel2122old 41042	. . . . 5 ⊢ ((Tr 𝐴 ∧ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
26	25	ex 415	. . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
27	26	alrimivv 1922	. . 3 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
28		dftr2 5165	. . . 4 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
29	28	biimpri 230	. . 3 ⊢ (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
30	27, 29	syl 17	. 2 ⊢ (Tr 𝐴 → Tr suc 𝐴)
31	30	iin1 40896	1 ⊢ (Tr 𝐴 → Tr suc 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843  ∀wal 1528   = wceq 1530   ∈ wcel 2107   ⊆ wss 3934  Tr wtr 5163  suc csuc 6186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939  df-in 3941  df-ss 3950  df-sn 4560  df-uni 4831  df-tr 5164  df-suc 6190

This theorem is referenced by: (None)