Theorem ondomon 4839

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 4405	. . . . . . . . . . . . 13 |- ((y ~<_ z /\ z ~<_ A) -> y ~<_ A)
2	1	anim2i 335	. . . . . . . . . . . 12 |- ((y e. On /\ (y ~<_ z /\ z ~<_ A)) -> (y e. On /\ y ~<_ A))
3	2	anassrs 441	. . . . . . . . . . 11 |- (((y e. On /\ y ~<_ z) /\ z ~<_ A) -> (y e. On /\ y ~<_ A))
4		onelon 2968	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y e. On)
5		onelsst 2996	. . . . . . . . . . . . . 14 |- (z e. On -> (y e. z -> y (_ z))
6	5	imp 350	. . . . . . . . . . . . 13 |- ((z e. On /\ y e. z) -> y (_ z)
7		visset 1810	. . . . . . . . . . . . . 14 |- y e. V
8		ssdomg 4398	. . . . . . . . . . . . . 14 |- (y e. V -> (y (_ z -> y ~<_ z))
9	7, 8	ax-mp 7	. . . . . . . . . . . . 13 |- (y (_ z -> y ~<_ z)
10	6, 9	syl 10	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y ~<_ z)
11	4, 10	jca 288	. . . . . . . . . . 11 |- ((z e. On /\ y e. z) -> (y e. On /\ y ~<_ z))
12	3, 11	sylan 448	. . . . . . . . . 10 |- (((z e. On /\ y e. z) /\ z ~<_ A) -> (y e. On /\ y ~<_ A))
13	12	exp31 376	. . . . . . . . 9 |- (z e. On -> (y e. z -> (z ~<_ A -> (y e. On /\ y ~<_ A))))
14	13	com12 11	. . . . . . . 8 |- (y e. z -> (z e. On -> (z ~<_ A -> (y e. On /\ y ~<_ A))))
15	14	imp3a 361	. . . . . . 7 |- (y e. z -> ((z e. On /\ z ~<_ A) -> (y e. On /\ y ~<_ A)))
16		breq1 2618	. . . . . . . 8 |- (x = z -> (x ~<_ A <-> z ~<_ A))
17	16	elrab 1902	. . . . . . 7 |- (z e. {x e. On | x ~<_ A} <-> (z e. On /\ z ~<_ A))
18		breq1 2618	. . . . . . . 8 |- (x = y -> (x ~<_ A <-> y ~<_ A))
19	18	elrab 1902	. . . . . . 7 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
20	15, 17, 19	3imtr4g 552	. . . . . 6 |- (y e. z -> (z e. {x e. On | x ~<_ A} -> y e. {x e. On | x ~<_ A}))
21	20	imp 350	. . . . 5 |- ((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
22	21	gen2 982	. . . 4 |- A.yA.z((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
23		dftr2 2678	. . . 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 190	. . 3 |- Tr {x e. On | x ~<_ A}
25		ssrab2 2128	. . 3 |- {x e. On | x ~<_ A} (_ On
26		ordon 2983	. . 3 |- Ord On
27		trssord 2961	. . 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 915	. 2 |- Ord {x e. On | x ~<_ A}
29		elisset 1814	. . . . 5 |- (A e. B -> A e. V)
30		domsdomtr 4465	. . . . . . . . 9 |- ((x ~<_ A /\ A ~< P~A) -> x ~< P~A)
31		canth2g 4474	. . . . . . . . 9 |- (A e. V -> A ~< P~A)
32	30, 31	sylan2 451	. . . . . . . 8 |- ((x ~<_ A /\ A e. V) -> x ~< P~A)
33	32	expcom 374	. . . . . . 7 |- (A e. V -> (x ~<_ A -> x ~< P~A))
34	33	a1d 12	. . . . . 6 |- (A e. V -> (x e. On -> (x ~<_ A -> x ~< P~A)))
35	34	r19.21aiv 1711	. . . . 5 |- (A e. V -> A.x e. On (x ~<_ A -> x ~< P~A))
36	29, 35	syl 10	. . . 4 |- (A e. B -> A.x e. On (x ~<_ A -> x ~< P~A))
37		ss2rab 2120	. . . 4 |- ({x e. On | x ~<_ A} (_ {x e. On | x ~< P~A} <-> A.x e. On (x ~<_ A -> x ~< P~A))
38	36, 37	sylibr 200	. . 3 |- (A e. B -> {x e. On | x ~<_ A} (_ {x e. On | x ~< P~A})
39		cardval2 4838	. . . . 5 |- (card` P~A) = {x e. On | x ~< P~A}
40		fvex 3727	. . . . 5 |- (card` P~A) e. V
41	39, 40	eqeltrr 1543	. . . 4 |- {x e. On | x ~< P~A} e. V
42	41	ssex 2715	. . 3 |- ({x e. On | x ~<_ A} (_ {x e. On | x ~< P~A} -> {x e. On | x ~<_ A} e. V)
43		elong 2952	. . 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 20	. 2 |- (A e. B -> ({x e. On | x ~<_ A} e. On <-> Ord {x e. On | x ~<_ A}))
45	28, 44	mpbiri 194	1 |- (A e. B -> {x e. On | x ~<_ A} e. On)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   e. wcel 957  A.wral 1643  {crab 1646  Vcvv 1808   (_ wss 2044  P~cpw 2398   class class class wbr 2615  Tr wtr 2676  Ord word 2943  Oncon0 2944  ` cfv 3178   ~<_ cdom 4358   ~< csdm 4359  cardccrd 4796

This theorem is referenced by:  ondomcard 4840

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-er 4254  df-en 4360  df-dom 4361  df-sdom 4362  df-card 4799

Copyright terms: Public domain