Theorem ordon 2238

Description: The class of all ordinal numbers is ordinal. Proposition 7.12 of [TakeutiZaring] p. 38, but without using the Axiom of Regularity.

Assertion

Ref	Expression
ordon	⊢ Ord On

Proof of Theorem ordon

Step	Hyp	Ref	Expression
1		dftr3 2045	. . . 4 ⊢ (Tr On ↔ ∀x ∈ On x ⊆ On)
2		ordelord 2221	. . . . . . . 8 ⊢ ((Ord x ∧ y ∈ x) → Ord y)
3		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
4	3	elon 2208	. . . . . . . 8 ⊢ (x ∈ On ↔ Ord x)
5	2, 4	sylanb 344	. . . . . . 7 ⊢ ((x ∈ On ∧ y ∈ x) → Ord y)
6	5	exp 291	. . . . . 6 ⊢ (x ∈ On → (y ∈ x → Ord y))
7		visset 1350	. . . . . . 7 ⊢ y ∈ V
8	7	elon 2208	. . . . . 6 ⊢ (y ∈ On ↔ Ord y)
9	6, 8	syl6ibr 186	. . . . 5 ⊢ (x ∈ On → (y ∈ x → y ∈ On))
10	9	ssrdv 1509	. . . 4 ⊢ (x ∈ On → x ⊆ On)
11	1, 10	mprgbir 1250	. . 3 ⊢ Tr On
12		onfr 2237	. . . . 5 ⊢ E Fr On
13		ordtri3or 2230	. . . . . . . 8 ⊢ ((Ord x ∧ Ord y) → (x ∈ y ∨ x = y ∨ y ∈ x))
14		epel 2124	. . . . . . . . 9 ⊢ (xEy ↔ x ∈ y)
15		pm4.2 148	. . . . . . . . 9 ⊢ (x = y ↔ x = y)
16		epel 2124	. . . . . . . . 9 ⊢ (yEx ↔ y ∈ x)
17	14, 15, 16	bi3or 607	. . . . . . . 8 ⊢ ((xEy ∨ x = y ∨ yEx) ↔ (x ∈ y ∨ x = y ∨ y ∈ x))
18	13, 17	sylibr 175	. . . . . . 7 ⊢ ((Ord x ∧ Ord y) → (xEy ∨ x = y ∨ yEx))
19		eloni 2209	. . . . . . 7 ⊢ (x ∈ On → Ord x)
20		eloni 2209	. . . . . . 7 ⊢ (y ∈ On → Ord y)
21	18, 19, 20	syl2an 349	. . . . . 6 ⊢ ((x ∈ On ∧ y ∈ On) → (xEy ∨ x = y ∨ yEx))
22	21	rgen2 1248	. . . . 5 ⊢ ∀x ∈ On ∀y ∈ On (xEy ∨ x = y ∨ yEx)
23	12, 22	pm3.2i 234	. . . 4 ⊢ (E Fr On ∧ ∀x ∈ On ∀y ∈ On (xEy ∨ x = y ∨ yEx))
24		dfwe2 2187	. . . 4 ⊢ (E We On ↔ (E Fr On ∧ ∀x ∈ On ∀y ∈ On (xEy ∨ x = y ∨ yEx)))
25	23, 24	mpbir 165	. . 3 ⊢ E We On
26	11, 25	pm3.2i 234	. 2 ⊢ (Tr On ∧ E We On)
27		df-ord 2202	. 2 ⊢ (Ord On ↔ (Tr On ∧ E We On))
28	26, 27	mpbir 165	1 ⊢ Ord On

Syntax hints:   ∧ wa 196   ∨ w3o 580   = weq 797   ∈ wel 803   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  Tr wtr 2041   class class class wbr 2054  Ecep 2056   Fr wfr 2061   We wwe 2062  Ord word 2198  Oncon0 2199

This theorem is referenced by:  epweon 2239  onprc 2240  ordeleqon 2241  ordsson 2242  tfi 2244  ssorduni 2249  onint 2261  suceloni 2314  limon 2342  onuninsuc 2356  ordom 2382  tfrlem8 2956  ondomon 3662

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

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