Theorem ondomon 6008

Description: The collection of ordinal numbers dominated by a set is an ordinal number. (In general, not all collections of ordinal numbers are ordinal.) Theorem 56 of [Suppes] p. 227.

Assertion

Ref	Expression
ondomon	|- (A e. B -> {x e. On | x ~<_ A} e. On)

Distinct variable group:   x,A

Proof of Theorem ondomon

Step	Hyp	Ref	Expression
1		domtr 5474	. . . . . . . . . . . . 13 |- ((y ~<_ z /\ z ~<_ A) -> y ~<_ A)
2	1	anim2i 362	. . . . . . . . . . . 12 |- ((y e. On /\ (y ~<_ z /\ z ~<_ A)) -> (y e. On /\ y ~<_ A))
3	2	anassrs 489	. . . . . . . . . . 11 |- (((y e. On /\ y ~<_ z) /\ z ~<_ A) -> (y e. On /\ y ~<_ A))
4		onelon 3683	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y e. On)
5		onelss 3705	. . . . . . . . . . . . . 14 |- (z e. On -> (y e. z -> y C_ z))
6	5	imp 377	. . . . . . . . . . . . 13 |- ((z e. On /\ y e. z) -> y C_ z)
7		visset 2295	. . . . . . . . . . . . . 14 |- y e. _V
8		ssdomg 5467	. . . . . . . . . . . . . 14 |- (y e. _V -> (y C_ z -> y ~<_ z))
9	7, 8	ax-mp 7	. . . . . . . . . . . . 13 |- (y C_ z -> y ~<_ z)
10	6, 9	syl 12	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y ~<_ z)
11	4, 10	jca 310	. . . . . . . . . . 11 |- ((z e. On /\ y e. z) -> (y e. On /\ y ~<_ z))
12	3, 11	sylan 497	. . . . . . . . . 10 |- (((z e. On /\ y e. z) /\ z ~<_ A) -> (y e. On /\ y ~<_ A))
13	12	exp31 407	. . . . . . . . 9 |- (z e. On -> (y e. z -> (z ~<_ A -> (y e. On /\ y ~<_ A))))
14	13	com12 14	. . . . . . . 8 |- (y e. z -> (z e. On -> (z ~<_ A -> (y e. On /\ y ~<_ A))))
15	14	imp3a 388	. . . . . . 7 |- (y e. z -> ((z e. On /\ z ~<_ A) -> (y e. On /\ y ~<_ A)))
16		breq1 3341	. . . . . . . 8 |- (x = z -> (x ~<_ A <-> z ~<_ A))
17	16	elrab 2414	. . . . . . 7 |- (z e. {x e. On | x ~<_ A} <-> (z e. On /\ z ~<_ A))
18		breq1 3341	. . . . . . . 8 |- (x = y -> (x ~<_ A <-> y ~<_ A))
19	18	elrab 2414	. . . . . . 7 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
20	15, 17, 19	3imtr4g 612	. . . . . 6 |- (y e. z -> (z e. {x e. On | x ~<_ A} -> y e. {x e. On | x ~<_ A}))
21	20	imp 377	. . . . 5 |- ((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
22	21	gen2 1329	. . . 4 |- A.yA.z((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
23		dftr2 3413	. . . 4 |- (Tr {x e. On | x ~<_ A} <-> A.yA.z((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A}))
24	22, 23	mpbir 207	. . 3 |- Tr {x e. On | x ~<_ A}
25		ssrab2 2692	. . 3 |- {x e. On | x ~<_ A} C_ On
26		ordon 3863	. . 3 |- Ord On
27		trssord 3675	. . 3 |- ((Tr {x e. On | x ~<_ A} /\ {x e. On | x ~<_ A} C_ On /\ Ord On) -> Ord {x e. On | x ~<_ A})
28	24, 25, 26, 27	mp3an 1191	. 2 |- Ord {x e. On | x ~<_ A}
29		elisset 2299	. . . . 5 |- (A e. B -> A e. _V)
30		domsdomtr 5539	. . . . . . . . 9 |- ((x ~<_ A /\ A ~< ~PA) -> x ~< ~PA)
31		canth2g 5549	. . . . . . . . 9 |- (A e. _V -> A ~< ~PA)
32	30, 31	sylan2 500	. . . . . . . 8 |- ((x ~<_ A /\ A e. _V) -> x ~< ~PA)
33	32	expcom 403	. . . . . . 7 |- (A e. _V -> (x ~<_ A -> x ~< ~PA))
34	33	a1d 15	. . . . . 6 |- (A e. _V -> (x e. On -> (x ~<_ A -> x ~< ~PA)))
35	34	r19.21aiv 2175	. . . . 5 |- (A e. _V -> A.x e. On (x ~<_ A -> x ~< ~PA))
36	29, 35	syl 12	. . . 4 |- (A e. B -> A.x e. On (x ~<_ A -> x ~< ~PA))
37		ss2rab 2683	. . . 4 |- ({x e. On | x ~<_ A} C_ {x e. On | x ~< ~PA} <-> A.x e. On (x ~<_ A -> x ~< ~PA))
38	36, 37	sylibr 217	. . 3 |- (A e. B -> {x e. On | x ~<_ A} C_ {x e. On | x ~< ~PA})
39		cardval2 6007	. . . . 5 |- (card` ~PA) = {x e. On | x ~< ~PA}
40		fvex 4689	. . . . 5 |- (card` ~PA) e. _V
41	39, 40	eqeltrri 1968	. . . 4 |- {x e. On | x ~< ~PA} e. _V
42	41	ssex 3455	. . 3 |- ({x e. On | x ~<_ A} C_ {x e. On | x ~< ~PA} -> {x e. On | x ~<_ A} e. _V)
43		elong 3665	. . 3 |- ({x e. On | x ~<_ A} e. _V -> ({x e. On | x ~<_ A} e. On <-> Ord {x e. On | x ~<_ A}))
44	38, 42, 43	3syl 24	. 2 |- (A e. B -> ({x e. On | x ~<_ A} e. On <-> Ord {x e. On | x ~<_ A}))
45	28, 44	mpbiri 211	1 |- (A e. B -> {x e. On | x ~<_ A} e. On)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032   class class class wbr 3338  Tr wtr 3411  Ord word 3656  Oncon0 3657  ` cfv 3998   ~<_ cdom 5424   ~< csdm 5425  cardccrd 5859

This theorem is referenced by:  ondomcard 6009

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862

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