Theorem suceloni 2314

Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41.

Assertion

Ref	Expression
suceloni	⊢ (A ∈ On → suc A ∈ On)

Proof of Theorem suceloni

Step	Hyp	Ref	Expression
1		ordon 2238	. . . 4 ⊢ Ord On
2		trssord 2216	. . . . 5 ⊢ ((Tr suc A ∧ suc A ⊆ On ∧ Ord On) → Ord suc A)
3	2	3exp 611	. . . 4 ⊢ (Tr suc A → (suc A ⊆ On → (Ord On → Ord suc A)))
4	1, 3	mpii 45	. . 3 ⊢ (Tr suc A → (suc A ⊆ On → Ord suc A))
5		onelsst 2255	. . . . . . . 8 ⊢ (A ∈ On → (x ∈ A → x ⊆ A))
6		elsn 1820	. . . . . . . . . 10 ⊢ (x ∈ {A} ↔ x = A)
7		eqimss 1548	. . . . . . . . . 10 ⊢ (x = A → x ⊆ A)
8	6, 7	sylbi 174	. . . . . . . . 9 ⊢ (x ∈ {A} → x ⊆ A)
9	8	a1i 7	. . . . . . . 8 ⊢ (A ∈ On → (x ∈ {A} → x ⊆ A))
10	5, 9	orim12d 436	. . . . . . 7 ⊢ (A ∈ On → ((x ∈ A ∨ x ∈ {A}) → (x ⊆ A ∨ x ⊆ A)))
11		df-suc 2205	. . . . . . . . 9 ⊢ suc A = (A ∪ {A})
12	11	eleq2i 1153	. . . . . . . 8 ⊢ (x ∈ suc A ↔ x ∈ (A ∪ {A}))
13		elun 1601	. . . . . . . 8 ⊢ (x ∈ (A ∪ {A}) ↔ (x ∈ A ∨ x ∈ {A}))
14	12, 13	bitr2 152	. . . . . . 7 ⊢ ((x ∈ A ∨ x ∈ {A}) ↔ x ∈ suc A)
15		oridm 208	. . . . . . 7 ⊢ ((x ⊆ A ∨ x ⊆ A) ↔ x ⊆ A)
16	10, 14, 15	3imtr3g 425	. . . . . 6 ⊢ (A ∈ On → (x ∈ suc A → x ⊆ A))
17		sssucid 2300	. . . . . . 7 ⊢ A ⊆ suc A
18		sstr2 1510	. . . . . . 7 ⊢ (x ⊆ A → (A ⊆ suc A → x ⊆ suc A))
19	17, 18	mpi 44	. . . . . 6 ⊢ (x ⊆ A → x ⊆ suc A)
20	16, 19	syl6 23	. . . . 5 ⊢ (A ∈ On → (x ∈ suc A → x ⊆ suc A))
21	20	r19.21aiv 1259	. . . 4 ⊢ (A ∈ On → ∀x ∈ suc Ax ⊆ suc A)
22		dftr3 2045	. . . 4 ⊢ (Tr suc A ↔ ∀x ∈ suc Ax ⊆ suc A)
23	21, 22	sylibr 175	. . 3 ⊢ (A ∈ On → Tr suc A)
24		onsst 2243	. . . . . 6 ⊢ (A ∈ On → A ⊆ On)
25		snssi 1851	. . . . . 6 ⊢ (A ∈ On → {A} ⊆ On)
26	24, 25	jca 236	. . . . 5 ⊢ (A ∈ On → (A ⊆ On ∧ {A} ⊆ On))
27		unss 1632	. . . . 5 ⊢ ((A ⊆ On ∧ {A} ⊆ On) ↔ (A ∪ {A}) ⊆ On)
28	26, 27	sylib 173	. . . 4 ⊢ (A ∈ On → (A ∪ {A}) ⊆ On)
29	28, 11	syl5ss 1544	. . 3 ⊢ (A ∈ On → suc A ⊆ On)
30	4, 23, 29	sylc 62	. 2 ⊢ (A ∈ On → Ord suc A)
31		sucexg 2302	. . 3 ⊢ (A ∈ On → suc A ∈ V)
32		elong 2207	. . 3 ⊢ (suc A ∈ V → (suc A ∈ On ↔ Ord suc A))
33	31, 32	syl 12	. 2 ⊢ (A ∈ On → (suc A ∈ On ↔ Ord suc A))
34	30, 33	mpbird 171	1 ⊢ (A ∈ On → suc A ∈ On)

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487  {csn 1808  Tr wtr 2041  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  onpwsuc 2315  ordsuc 2318  unon 2338  onsuc 2353  limsuc 2361  ordzsl 2366  dfom2 2374  findsg 2398  tfindsg 2402  tfrlem12 2960  oasuc 3131  omsuc 3133  oesuc 3134  oacl 3138  nnacom 3175  r1ord 3499  rankwflem 3509  rankr1 3518  bndrank 3526  r1pw 3529

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

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