Theorem ondomon 3662

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(s):   x,A

Proof of Theorem ondomon

Step	Hyp	Ref	Expression
1		domtr 3320	. . . . . . . . . . . . 13 |- ((y ~<_ z /\ z ~<_ A) -> y ~<_ A)
2	1	anim2i 270	. . . . . . . . . . . 12 |- ((y e. On /\ (y ~<_ z /\ z ~<_ A)) -> (y e. On /\ y ~<_ A))
3	2	anassrs 338	. . . . . . . . . . 11 |- (((y e. On /\ y ~<_ z) /\ z ~<_ A) -> (y e. On /\ y ~<_ A))
4		onelon 2223	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y e. On)
5		onelsst 2255	. . . . . . . . . . . . . 14 |- (z e. On -> (y e. z -> y (_ z))
6	5	imp 277	. . . . . . . . . . . . 13 |- ((z e. On /\ y e. z) -> y (_ z)
7		visset 1350	. . . . . . . . . . . . . 14 |- y e. V
8		ssdomg 3311	. . . . . . . . . . . . . 14 |- (y e. V -> (y (_ z -> y ~<_ z))
9	7, 8	ax-mp 6	. . . . . . . . . . . . 13 |- (y (_ z -> y ~<_ z)
10	6, 9	syl 12	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y ~<_ z)
11	4, 10	jca 236	. . . . . . . . . . 11 |- ((z e. On /\ y e. z) -> (y e. On /\ y ~<_ z))
12	3, 11	sylan 343	. . . . . . . . . 10 |- (((z e. On /\ y e. z) /\ z ~<_ A) -> (y e. On /\ y ~<_ A))
13	12	exp31 293	. . . . . . . . 9 |- (z e. On -> (y e. z -> (z ~<_ A -> (y e. On /\ y ~<_ A))))
14	13	com12 13	. . . . . . . 8 |- (y e. z -> (z e. On -> (z ~<_ A -> (y e. On /\ y ~<_ A))))
15	14	imp3a 279	. . . . . . 7 |- (y e. z -> ((z e. On /\ z ~<_ A) -> (y e. On /\ y ~<_ A)))
16		breq1 2065	. . . . . . . 8 |- (x = z -> (x ~<_ A <-> z ~<_ A))
17	16	elrab 1422	. . . . . . 7 |- (z e. {x e. On | x ~<_ A} <-> (z e. On /\ z ~<_ A))
18		breq1 2065	. . . . . . . 8 |- (x = y -> (x ~<_ A <-> y ~<_ A))
19	18	elrab 1422	. . . . . . 7 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
20	15, 17, 19	3imtr4g 426	. . . . . 6 |- (y e. z -> (z e. {x e. On | x ~<_ A} -> y e. {x e. On | x ~<_ A}))
21	20	imp 277	. . . . 5 |- ((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
22	21	gen2 681	. . . 4 |- A.yA.z((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
23		dftr2 2043	. . . 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 165	. . 3 |- Tr {x e. On | x ~<_ A}
25		ssrab 1556	. . 3 |- {x e. On | x ~<_ A} (_ On
26		ordon 2238	. . 3 |- Ord On
27		trssord 2216	. . 3 |- ((Tr {x e. On | x ~<_ A} /\ {x e. On | x ~<_ A} (_ On /\ Ord On) -> Ord {x e. On | x ~<_ A})
28	24, 25, 26, 27	mp3an 642	. 2 |- Ord {x e. On | x ~<_ A}
29		elisset 1354	. . . . 5 |- (A e. B -> A e. V)
30		domsdomtr 3374	. . . . . . . . . 10 |- ((x ~<_ A /\ A ~< P~A) -> x ~< P~A)
31		canth2g 3382	. . . . . . . . . 10 |- (A e. V -> A ~< P~A)
32	30, 31	sylan2 346	. . . . . . . . 9 |- ((x ~<_ A /\ A e. V) -> x ~< P~A)
33	32	exp 291	. . . . . . . 8 |- (x ~<_ A -> (A e. V -> x ~< P~A))
34	33	com12 13	. . . . . . 7 |- (A e. V -> (x ~<_ A -> x ~< P~A))
35	34	a1d 14	. . . . . 6 |- (A e. V -> (x e. On -> (x ~<_ A -> x ~< P~A)))
36	35	r19.21aiv 1259	. . . . 5 |- (A e. V -> A.x e. On (x ~<_ A -> x ~< P~A))
37	29, 36	syl 12	. . . 4 |- (A e. B -> A.x e. On (x ~<_ A -> x ~< P~A))
38		ss2rab 1553	. . . 4 |- ({x e. On | x ~<_ A} (_ {x e. On | x ~< P~A} <-> A.x e. On (x ~<_ A -> x ~< P~A))
39	37, 38	sylibr 175	. . 3 |- (A e. B -> {x e. On | x ~<_ A} (_ {x e. On | x ~< P~A})
40		cardval2 3661	. . . . 5 |- (card` P~A) = {x e. On | x ~< P~A}
41		fvex 2838	. . . . 5 |- (card` P~A) e. V
42	40, 41	eqeltrr 1160	. . . 4 |- {x e. On | x ~< P~A} e. V
43	42	ssex 1700	. . 3 |- ({x e. On | x ~<_ A} (_ {x e. On | x ~< P~A} -> {x e. On | x ~<_ A} e. V)
44		elong 2207	. . 3 |- ({x e. On | x ~<_ A} e. V -> ({x e. On | x ~<_ A} e. On <-> Ord {x e. On | x ~<_ A}))
45	39, 43, 44	3syl 21	. 2 |- (A e. B -> ({x e. On | x ~<_ A} e. On <-> Ord {x e. On | x ~<_ A}))
46	28, 45	mpbiri 169	1 |- (A e. B -> {x e. On | x ~<_ A} e. On)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wel 803   e. wcel 1092  A.wral 1201  {crab 1204  Vcvv 1348   (_ wss 1487  P~cpw 1798  Tr wtr 2041   class class class wbr 2054  Ord word 2198  Oncon0 2199  ` cfv 2422   ~<_ cdom 3272   ~< csdm 3273  cardccrd 3620

This theorem is referenced by:  ondomcard 3663

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  ax-reg 1078  ax-ac 1080

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-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  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-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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