Theorem ordon 2977

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		df-ord 2941	. 2 |- (Ord On <-> (Tr On /\ E We On))
2		dftr3 2674	. . 3 |- (Tr On <-> A.x e. On x (_ On)
3		ordelord 2960	. . . . . . 7 |- ((Ord x /\ y e. x) -> Ord y)
4		visset 1804	. . . . . . . 8 |- x e. V
5	4	elon 2947	. . . . . . 7 |- (x e. On <-> Ord x)
6	3, 5	sylanb 449	. . . . . 6 |- ((x e. On /\ y e. x) -> Ord y)
7	6	ex 373	. . . . 5 |- (x e. On -> (y e. x -> Ord y))
8		visset 1804	. . . . . 6 |- y e. V
9	8	elon 2947	. . . . 5 |- (y e. On <-> Ord y)
10	7, 9	syl6ibr 213	. . . 4 |- (x e. On -> (y e. x -> y e. On))
11	10	ssrdv 2060	. . 3 |- (x e. On -> x (_ On)
12	2, 11	mprgbir 1693	. 2 |- Tr On
13		dfwe2 2925	. . 3 |- (E We On <-> (E Fr On /\ A.x e. On A.y e. On (xEy \/ x = y \/ yEx)))
14		onfr 2976	. . 3 |- E Fr On
15		ordtri3or 2969	. . . . . 6 |- ((Ord x /\ Ord y) -> (x e. y \/ x = y \/ y e. x))
16		epel 2823	. . . . . . 7 |- (xEy <-> x e. y)
17		pm4.2 170	. . . . . . 7 |- (x = y <-> x = y)
18		epel 2823	. . . . . . 7 |- (yEx <-> y e. x)
19	16, 17, 18	3orbi123i 821	. . . . . 6 |- ((xEy \/ x = y \/ yEx) <-> (x e. y \/ x = y \/ y e. x))
20	15, 19	sylibr 200	. . . . 5 |- ((Ord x /\ Ord y) -> (xEy \/ x = y \/ yEx))
21		eloni 2948	. . . . 5 |- (x e. On -> Ord x)
22		eloni 2948	. . . . 5 |- (y e. On -> Ord y)
23	20, 21, 22	syl2an 454	. . . 4 |- ((x e. On /\ y e. On) -> (xEy \/ x = y \/ yEx))
24	23	rgen2a 1691	. . 3 |- A.x e. On A.y e. On (xEy \/ x = y \/ yEx)
25	13, 14, 24	mpbir2an 728	. 2 |- E We On
26	1, 12, 25	mpbir2an 728	1 |- Ord On

Syntax hints:   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  A.wral 1637   (_ wss 2037   class class class wbr 2609  Tr wtr 2670  Ecep 2819   Fr wfr 2905   We wwe 2906  Ord word 2937  Oncon0 2938

This theorem is referenced by:  epweon 2978  onprc 2979  ordeleqon 2980  ordsson 2981  ssorduni 2983  onint 2996  suceloni 3052  limon 3084  onuninsuc 3098  tfi 3116  ordom 3131  ondomon 4828

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942

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