Theorem ondomon 4856

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 4415	. . . . . . . . . . . . 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 2972	. . . . . . . . . . . 12 |- ((z e. On /\ y e. z) -> y e. On)
5		onelsst 3000	. . . . . . . . . . . . . 14 |- (z e. On -> (y e. z -> y (_ z))
6	5	imp 350	. . . . . . . . . . . . 13 |- ((z e. On /\ y e. z) -> y (_ z)
7		visset 1813	. . . . . . . . . . . . . 14 |- y e. V
8		ssdomg 4408	. . . . . . . . . . . . . 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 2622	. . . . . . . 8 |- (x = z -> (x ~<_ A <-> z ~<_ A))
17	16	elrab 1905	. . . . . . 7 |- (z e. {x e. On | x ~<_ A} <-> (z e. On /\ z ~<_ A))
18		breq1 2622	. . . . . . . 8 |- (x = y -> (x ~<_ A <-> y ~<_ A))
19	18	elrab 1905	. . . . . . 7 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
20	15, 17, 19	3imtr4g 553	. . . . . 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 983	. . . 4 |- A.yA.z((y e. z /\ z e. {x e. On | x ~<_ A}) -> y e. {x e. On | x ~<_ A})
23		dftr2 2682	. . . 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 2131	. . 3 |- {x e. On | x ~<_ A} (_ On
26		ordon 2987	. . 3 |- Ord On
27		trssord 2965	. . 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 916	. 2 |- Ord {x e. On | x ~<_ A}
29		elisset 1817	. . . . 5 |- (A e. B -> A e. V)
30		domsdomtr 4476	. . . . . . . . 9 |- ((x ~<_ A /\ A ~< P~A) -> x ~< P~A)
31		canth2g 4485	. . . . . . . . 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 1713	. . . . 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 2123	. . . 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 4855	. . . . 5 |- (card` P~A) = {x e. On | x ~< P~A}
40		fvex 3732	. . . . 5 |- (card` P~A) e. V
41	39, 40	eqeltrr 1545	. . . 4 |- {x e. On | x ~< P~A} e. V
42	41	ssex 2719	. . 3 |- ({x e. On | x ~<_ A} (_ {x e. On | x ~< P~A} -> {x e. On | x ~<_ A} e. V)
43		elong 2956	. . 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 954   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811   (_ wss 2047  P~cpw 2401   class class class wbr 2619  Tr wtr 2680  Ord word 2947  Oncon0 2948  ` cfv 3182   ~<_ cdom 4365   ~< csdm 4366  cardccrd 4813

This theorem is referenced by:  ondomcard 4857

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816

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