Theorem ondomcard 4857

Description: The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228.

Assertion

Ref	Expression
ondomcard	|- (A e. B -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})

Distinct variable group:   x,A

Proof of Theorem ondomcard

Step	Hyp	Ref	Expression
1		elisset 1817	. 2 |- (A e. B -> A e. V)
2		ondomon 4856	. . . 4 |- (A e. V -> {x e. On | x ~<_ A} e. On)
3		domsdomtr 4476	. . . . . . . . . . . 12 |- ((y ~<_ A /\ A ~< {x e. On | x ~<_ A}) -> y ~< {x e. On | x ~<_ A})
4		breq1 2622	. . . . . . . . . . . . . 14 |- (x = y -> (x ~<_ A <-> y ~<_ A))
5	4	elrab 1905	. . . . . . . . . . . . 13 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
6	5	pm3.27bi 326	. . . . . . . . . . . 12 |- (y e. {x e. On | x ~<_ A} -> y ~<_ A)
7		eloni 2958	. . . . . . . . . . . . . . . 16 |- ({x e. On | x ~<_ A} e. On -> Ord {x e. On | x ~<_ A})
8		ordirr 2966	. . . . . . . . . . . . . . . 16 |- (Ord {x e. On | x ~<_ A} -> -. {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
9	7, 8	syl 10	. . . . . . . . . . . . . . 15 |- ({x e. On | x ~<_ A} e. On -> -. {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
10		hbrab1 1772	. . . . . . . . . . . . . . . . . 18 |- (y e. {x e. On | x ~<_ A} -> A.x y e. {x e. On | x ~<_ A})
11		ax-17 971	. . . . . . . . . . . . . . . . . 18 |- (y e. On -> A.x y e. On)
12		ax-17 971	. . . . . . . . . . . . . . . . . . 19 |- (y e. ~<_ -> A.x y e. ~<_ )
13		ax-17 971	. . . . . . . . . . . . . . . . . . 19 |- (y e. A -> A.x y e. A)
14	10, 12, 13	hbbr 2658	. . . . . . . . . . . . . . . . . 18 |- ({x e. On | x ~<_ A} ~<_ A -> A.x{x e. On | x ~<_ A} ~<_ A)
15		breq1 2622	. . . . . . . . . . . . . . . . . 18 |- (x = {x e. On | x ~<_ A} -> (x ~<_ A <-> {x e. On | x ~<_ A} ~<_ A))
16	10, 11, 14, 15	elrabf 1904	. . . . . . . . . . . . . . . . 17 |- ({x e. On | x ~<_ A} e. {x e. On | x ~<_ A} <-> ({x e. On | x ~<_ A} e. On /\ {x e. On | x ~<_ A} ~<_ A))
17	16	biimpr 152	. . . . . . . . . . . . . . . 16 |- (({x e. On | x ~<_ A} e. On /\ {x e. On | x ~<_ A} ~<_ A) -> {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
18	17	ex 373	. . . . . . . . . . . . . . 15 |- ({x e. On | x ~<_ A} e. On -> ({x e. On | x ~<_ A} ~<_ A -> {x e. On | x ~<_ A} e. {x e. On | x ~<_ A}))
19	9, 18	mtod 108	. . . . . . . . . . . . . 14 |- ({x e. On | x ~<_ A} e. On -> -. {x e. On | x ~<_ A} ~<_ A)
20	2, 19	syl 10	. . . . . . . . . . . . 13 |- (A e. V -> -. {x e. On | x ~<_ A} ~<_ A)
21		domtri 4838	. . . . . . . . . . . . . . 15 |- (({x e. On | x ~<_ A} e. On /\ A e. V) -> ({x e. On | x ~<_ A} ~<_ A <-> -. A ~< {x e. On | x ~<_ A}))
22	21	con2bid 526	. . . . . . . . . . . . . 14 |- (({x e. On | x ~<_ A} e. On /\ A e. V) -> (A ~< {x e. On | x ~<_ A} <-> -. {x e. On | x ~<_ A} ~<_ A))
23	2, 22	mpancom 705	. . . . . . . . . . . . 13 |- (A e. V -> (A ~< {x e. On | x ~<_ A} <-> -. {x e. On | x ~<_ A} ~<_ A))
24	20, 23	mpbird 196	. . . . . . . . . . . 12 |- (A e. V -> A ~< {x e. On | x ~<_ A})
25	3, 6, 24	syl2an 454	. . . . . . . . . . 11 |- ((y e. {x e. On | x ~<_ A} /\ A e. V) -> y ~< {x e. On | x ~<_ A})
26		sdomnen 4387	. . . . . . . . . . 11 |- (y ~< {x e. On | x ~<_ A} -> -. y ~~ {x e. On | x ~<_ A})
27	25, 26	syl 10	. . . . . . . . . 10 |- ((y e. {x e. On | x ~<_ A} /\ A e. V) -> -. y ~~ {x e. On | x ~<_ A})
28	27	expcom 374	. . . . . . . . 9 |- (A e. V -> (y e. {x e. On | x ~<_ A} -> -. y ~~ {x e. On | x ~<_ A}))
29	28	con2d 91	. . . . . . . 8 |- (A e. V -> (y ~~ {x e. On | x ~<_ A} -> -. y e. {x e. On | x ~<_ A}))
30		visset 1813	. . . . . . . . 9 |- y e. V
31	30	ensym 4412	. . . . . . . 8 |- ({x e. On | x ~<_ A} ~~ y -> y ~~ {x e. On | x ~<_ A})
32	29, 31	syl5 21	. . . . . . 7 |- (A e. V -> ({x e. On | x ~<_ A} ~~ y -> -. y e. {x e. On | x ~<_ A}))
33	32	adantr 389	. . . . . 6 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> -. y e. {x e. On | x ~<_ A}))
34		ontri1 2981	. . . . . . 7 |- (({x e. On | x ~<_ A} e. On /\ y e. On) -> ({x e. On | x ~<_ A} (_ y <-> -. y e. {x e. On | x ~<_ A}))
35	34, 2	sylan 448	. . . . . 6 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} (_ y <-> -. y e. {x e. On | x ~<_ A}))
36	33, 35	sylibrd 204	. . . . 5 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y))
37	36	r19.21aiva 1714	. . . 4 |- (A e. V -> A.y e. On ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y))
38	2, 37	jca 288	. . 3 |- (A e. V -> ({x e. On | x ~<_ A} e. On /\ A.y e. On ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y)))
39		iscard2 4854	. . 3 |- ((card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A} <-> ({x e. On | x ~<_ A} e. On /\ A.y e. On ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y)))
40	38, 39	sylibr 200	. 2 |- (A e. V -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})
41	1, 40	syl 10	1 |- (A e. B -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811   (_ wss 2047   class class class wbr 2619  Ord word 2947  Oncon0 2948  ` cfv 3182   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366  cardccrd 4813

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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