Theorem ondomcard 6009

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 2299	. 2 |- (A e. B -> A e. _V)
2		iscard2 6006	. . 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} C_ y)))
3		ondomon 6008	. . 3 |- (A e. _V -> {x e. On | x ~<_ A} e. On)
4		domsdomtr 5539	. . . . . . . . . . 11 |- ((y ~<_ A /\ A ~< {x e. On | x ~<_ A}) -> y ~< {x e. On | x ~<_ A})
5		breq1 3341	. . . . . . . . . . . . 13 |- (x = y -> (x ~<_ A <-> y ~<_ A))
6	5	elrab 2414	. . . . . . . . . . . 12 |- (y e. {x e. On | x ~<_ A} <-> (y e. On /\ y ~<_ A))
7	6	simprbi 353	. . . . . . . . . . 11 |- (y e. {x e. On | x ~<_ A} -> y ~<_ A)
8		eloni 3667	. . . . . . . . . . . . . . 15 |- ({x e. On | x ~<_ A} e. On -> Ord {x e. On | x ~<_ A})
9		ordirr 3676	. . . . . . . . . . . . . . 15 |- (Ord {x e. On | x ~<_ A} -> -. {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
10	8, 9	syl 12	. . . . . . . . . . . . . 14 |- ({x e. On | x ~<_ A} e. On -> -. {x e. On | x ~<_ A} e. {x e. On | x ~<_ A})
11		hbrab1 2257	. . . . . . . . . . . . . . . 16 |- (y e. {x e. On | x ~<_ A} -> A.x y e. {x e. On | x ~<_ A})
12		ax-17 1317	. . . . . . . . . . . . . . . 16 |- (y e. On -> A.x y e. On)
13		ax-17 1317	. . . . . . . . . . . . . . . . 17 |- (y e. ~<_ -> A.x y e. ~<_ )
14		ax-17 1317	. . . . . . . . . . . . . . . . 17 |- (y e. A -> A.x y e. A)
15	11, 13, 14	hbbr 3381	. . . . . . . . . . . . . . . 16 |- ({x e. On | x ~<_ A} ~<_ A -> A.x{x e. On | x ~<_ A} ~<_ A)
16		breq1 3341	. . . . . . . . . . . . . . . 16 |- (x = {x e. On | x ~<_ A} -> (x ~<_ A <-> {x e. On | x ~<_ A} ~<_ A))
17	11, 12, 15, 16	elrabf 2413	. . . . . . . . . . . . . . 15 |- ({x e. On | x ~<_ A} e. {x e. On | x ~<_ A} <-> ({x e. On | x ~<_ A} e. On /\ {x e. On | x ~<_ A} ~<_ A))
18	17	biimpri 169	. . . . . . . . . . . . . 14 |- (({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	10, 18	mtand 520	. . . . . . . . . . . . 13 |- ({x e. On | x ~<_ A} e. On -> -. {x e. On | x ~<_ A} ~<_ A)
20	3, 19	syl 12	. . . . . . . . . . . 12 |- (A e. _V -> -. {x e. On | x ~<_ A} ~<_ A)
21		domtri 5989	. . . . . . . . . . . . . 14 |- (({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 585	. . . . . . . . . . . . 13 |- (({x e. On | x ~<_ A} e. On /\ A e. _V) -> (A ~< {x e. On | x ~<_ A} <-> -. {x e. On | x ~<_ A} ~<_ A))
23	3, 22	mpancom 769	. . . . . . . . . . . 12 |- (A e. _V -> (A ~< {x e. On | x ~<_ A} <-> -. {x e. On | x ~<_ A} ~<_ A))
24	20, 23	mpbird 213	. . . . . . . . . . 11 |- (A e. _V -> A ~< {x e. On | x ~<_ A})
25	4, 7, 24	syl2an 503	. . . . . . . . . 10 |- ((y e. {x e. On | x ~<_ A} /\ A e. _V) -> y ~< {x e. On | x ~<_ A})
26		sdomnen 5446	. . . . . . . . . 10 |- (y ~< {x e. On | x ~<_ A} -> -. y ~~ {x e. On | x ~<_ A})
27	25, 26	syl 12	. . . . . . . . 9 |- ((y e. {x e. On | x ~<_ A} /\ A e. _V) -> -. y ~~ {x e. On | x ~<_ A})
28	27	expcom 403	. . . . . . . 8 |- (A e. _V -> (y e. {x e. On | x ~<_ A} -> -. y ~~ {x e. On | x ~<_ A}))
29	28	con2d 107	. . . . . . 7 |- (A e. _V -> (y ~~ {x e. On | x ~<_ A} -> -. y e. {x e. On | x ~<_ A}))
30		visset 2295	. . . . . . . 8 |- y e. _V
31	30	ensym 5471	. . . . . . 7 |- ({x e. On | x ~<_ A} ~~ y -> y ~~ {x e. On | x ~<_ A})
32	29, 31	syl5 20	. . . . . 6 |- (A e. _V -> ({x e. On | x ~<_ A} ~~ y -> -. y e. {x e. On | x ~<_ A}))
33	32	adantr 425	. . . . 5 |- ((A e. _V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> -. y e. {x e. On | x ~<_ A}))
34		ontri1 3695	. . . . . 6 |- (({x e. On | x ~<_ A} e. On /\ y e. On) -> ({x e. On | x ~<_ A} C_ y <-> -. y e. {x e. On | x ~<_ A}))
35	34, 3	sylan 497	. . . . 5 |- ((A e. _V /\ y e. On) -> ({x e. On | x ~<_ A} C_ y <-> -. y e. {x e. On | x ~<_ A}))
36	33, 35	sylibrd 221	. . . 4 |- ((A e. _V /\ y e. On) -> ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} C_ y))
37	36	r19.21aiva 2176	. . 3 |- (A e. _V -> A.y e. On ({x e. On | x ~<_ A} ~~ y -> {x e. On | x ~<_ A} C_ y))
38	2, 3, 37	sylanbrc 527	. 2 |- (A e. _V -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})
39	1, 38	syl 12	1 |- (A e. B -> (card` {x e. On | x ~<_ A}) = {x e. On | x ~<_ A})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  Ord word 3656  Oncon0 3657  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  cardccrd 5859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862

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