Theorem ondomcard 5007

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 1863	. 2 |- (A e. B -> A e. V)
2		ondomon 5006	. . . 4 |- (A e. V -> {x e. On | x ~<_ A} e. On)
3		domsdomtr 4621	. . . . . . . . . . . 12 |- ((y ~<_ A /\ A ~< {x e. On | x ~<_ A}) -> y ~< {x e. On | x ~<_ A})
4		breq1 2695	. . . . . . . . . . . . . 14 |- (x = y -> (x ~<_ A <-> y ~<_ A))
5	4	elrab 1951	. . . . . . . . . . . . 13 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
6	5	pm3.27bi 324	. . . . . . . . . . . 12 |- (y e. {x e. On | x ~<_ A} -> y ~<_ A)
7		eloni 2985	. . . . . . . . . . . . . . . 16 |- ({x e. On | x ~<_ A} e. On -> Ord {x e. On | x ~<_ A})
8		ordirr 2993	. . . . . . . . . . . . . . . 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 1818	. . . . . . . . . . . . . . . . . 18 |- (y e. {x e. On | x ~<_ A} -> A.x y e. {x e. On | x ~<_ A})
11		ax-17 1007	. . . . . . . . . . . . . . . . . 18 |- (y e. On -> A.x y e. On)
12		ax-17 1007	. . . . . . . . . . . . . . . . . . 19 |- (y e. ~<_ -> A.x y e. ~<_ )
13		ax-17 1007	. . . . . . . . . . . . . . . . . . 19 |- (y e. A -> A.x y e. A)
14	10, 12, 13	hbbr 2731	. . . . . . . . . . . . . . . . . 18 |- ({x e. On | x ~<_ A} ~<_ A -> A.x{x e. On | x ~<_ A} ~<_ A)
15		breq1 2695	. . . . . . . . . . . . . . . . . 18 |- (x = {x e. On | x ~<_ A} -> (x ~<_ A <-> {x e. On | x ~<_ A} ~<_ A))
16	10, 11, 14, 15	elrabf 1950	. . . . . . . . . . . . . . . . 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	biimpri 150	. . . . . . . . . . . . . . . 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 371	. . . . . . . . . . . . . . 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 107	. . . . . . . . . . . . . 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 4987	. . . . . . . . . . . . . . 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 529	. . . . . . . . . . . . . 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 709	. . . . . . . . . . . . 13 |- (A e. V -> (A ~< {x e. On | x ~<_ A} <-> -. {x e. On | x ~<_ A} ~<_ A))
24	20, 23	mpbird 194	. . . . . . . . . . . 12 |- (A e. V -> A ~< {x e. On | x ~<_ A})
25	3, 6, 24	syl2an 456	. . . . . . . . . . 11 |- ((y e. {x e. On | x ~<_ A} /\ A e. V) -> y ~< {x e. On | x ~<_ A})
26		sdomnen 4528	. . . . . . . . . . 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 372	. . . . . . . . 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 1859	. . . . . . . . 9 |- y e. V
31	30	ensym 4553	. . . . . . . 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 3009	. . . . . . 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 450	. . . . . 6 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} (_ y <-> -. y e. {x e. On | x ~<_ A}))
36	33, 35	sylibrd 202	. . . . 5 |- ((A e. V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y))
37	36	r19.21aiva 1760	. . . 4 |- (A e. V -> A.y e. On ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} (_ y))
38	2, 37	jca 286	. . 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 5004	. . 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 198	. 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 144   /\ wa 221   = wceq 992   e. wcel 994  A.wral 1691  {crab 1694  Vcvv 1857   (_ wss 2099   class class class wbr 2692  Ord word 2974  Oncon0 2975  ` cfv 3263   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507  cardccrd 4959

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-suc 2981  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-card 4962

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