Theorem ondomcard 3663

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(s):   x,A

Proof of Theorem ondomcard

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 |- (A e. B -> A e. V)
2		ondomon 3662	. . . 4 |- (A e. V -> {x e. On | x ~<_ A} e. On)
3		domsdomtr 3374	. . . . . . . . . . . . . 14 |- ((y ~<_ A /\ A ~< {x e. On | x ~<_ A}) -> y ~< {x e. On | x ~<_ A})
4		breq1 2065	. . . . . . . . . . . . . . . 16 |- (x = y -> (x ~<_ A <-> y ~<_ A))
5	4	elrab 1422	. . . . . . . . . . . . . . 15 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
6	5	pm3.27bd 263	. . . . . . . . . . . . . 14 |- (y e. {x e. On | x ~<_ A} -> y ~<_ A)
7		eloni 2209	. . . . . . . . . . . . . . . . . 18 |- ({x e. On | x ~<_ A} e. On -> Ord {x e. On | x ~<_ A})
8		ordeirr 2217	. . . . . . . . . . . . . . . . . 18 |- (Ord {x e. On | x ~<_ A} -> -. {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
9	7, 8	syl 12	. . . . . . . . . . . . . . . . 17 |- ({x e. On | x ~<_ A} e. On -> -. {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
10		hbrab1 1310	. . . . . . . . . . . . . . . . . . . 20 |- (y e. {x e. On | x ~<_ A} -> A.x y e. {x e. On | x ~<_ A})
11		ax-17 925	. . . . . . . . . . . . . . . . . . . 20 |- (y e. On -> A.x y e. On)
12		ax-17 925	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. ~<_ -> A.x y e. ~<_ )
13		ax-17 925	. . . . . . . . . . . . . . . . . . . . 21 |- (y e. A -> A.x y e. A)
14	10, 12, 13	hbbr 2095	. . . . . . . . . . . . . . . . . . . 20 |- ({x e. On | x ~<_ A} ~<_ A -> A.x{x e. On | x ~<_ A} ~<_ A)
15		breq1 2065	. . . . . . . . . . . . . . . . . . . 20 |- (x = {x e. On | x ~<_ A} -> (x ~<_ A <-> {x e. On | x ~<_ A} ~<_ A))
16	10, 11, 14, 15	elrabf 1421	. . . . . . . . . . . . . . . . . . 19 |- ({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 134	. . . . . . . . . . . . . . . . . 18 |- (({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	exp 291	. . . . . . . . . . . . . . . . 17 |- ({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 95	. . . . . . . . . . . . . . . 16 |- ({x e. On | x ~<_ A} e. On -> -. {x e. On | x ~<_ A} ~<_ A)
20	2, 19	syl 12	. . . . . . . . . . . . . . 15 |- (A e. V -> -. {x e. On | x ~<_ A} ~<_ A)
21		domtri 3644	. . . . . . . . . . . . . . . . 17 |- (({x e. On | x ~<_ A} e. On /\ A e. V) -> ({x e. On | x ~<_ A} ~<_ A <-> -. A ~< {x e. On | x ~<_ A}))
22	21	bicon2d 404	. . . . . . . . . . . . . . . 16 |- (({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 528	. . . . . . . . . . . . . . 15 |- (A e. V -> (A ~< {x e. On | x ~<_ A} <-> -. {x e. On | x ~<_ A} ~<_ A))
24	20, 23	mpbird 171	. . . . . . . . . . . . . 14 |- (A e. V -> A ~< {x e. On | x ~<_ A})
25	3, 6, 24	syl2an 349	. . . . . . . . . . . . 13 |- ((y e. {x e. On | x ~<_ A} /\ A e. V) -> y ~< {x e. On | x ~<_ A})
26		sdomnen 3291	. . . . . . . . . . . . 13 |- (y ~< {x e. On | x ~<_ A} -> -. y ~~ {x e. On | x ~<_ A})
27	25, 26	syl 12	. . . . . . . . . . . 12 |- ((y e. {x e. On | x ~<_ A} /\ A e. V) -> -. y ~~ {x e. On | x ~<_ A})
28	27	exp 291	. . . . . . . . . . 11 |- (y e. {x e. On | x ~<_ A} -> (A e. V -> -. y ~~ {x e. On | x ~<_ A}))
29	28	com12 13	. . . . . . . . . 10 |- (A e. V -> (y e. {x e. On | x ~<_ A} -> -. y ~~ {x e. On | x ~<_ A}))
30	29	con2d 83	. . . . . . . . 9 |- (A e. V -> (y ~~ {x e. On | x ~<_ A} -> -. y e. {x e. On | x ~<_ A}))
31		visset 1350	. . . . . . . . . 10 |- y e. V
32	31	ensym 3317	. . . . . . . . 9 |- ({x e. On | x ~<_ A} ~~ y -> y ~~ {x e. On | x ~<_ A})
33	30, 32	syl5 22	. . . . . . . 8 |- (A e. V -> ({x e. On | x ~<_ A} ~~ y -> -. y e. {x e. On | x ~<_ A}))
34	33	adantr 306	. . . . . . 7 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> -. y e. {x e. On | x ~<_ A}))
35		ontri1 2232	. . . . . . . 8 |- (({x e. On | x ~<_ A} e. On /\ y e. On) -> ({x e. On | x ~<_ A} (_ y <-> -. y e. {x e. On | x ~<_ A}))
36	35, 2	sylan 343	. . . . . . 7 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} (_ y <-> -. y e. {x e. On | x ~<_ A}))
37	34, 36	sylibrd 179	. . . . . 6 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y))
38	37	exp 291	. . . . 5 |- (A e. V -> (y e. On -> ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y)))
39	38	r19.21aiv 1259	. . . 4 |- (A e. V -> A.y e. On ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y))
40	2, 39	jca 236	. . 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)))
41		iscard2 3660	. . 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)))
42	40, 41	sylibr 175	. 2 |- (A e. V -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})
43	1, 42	syl 12	1 |- (A e. B -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204  Vcvv 1348   (_ wss 1487   class class class wbr 2054  Ord word 2198  Oncon0 2199  ` cfv 2422   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273  cardccrd 3620

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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