Theorem harcard 8804

Description: The class of ordinal numbers dominated by a set is a cardinal number. Theorem 59 of [Suppes] p. 228. (Contributed by Mario Carneiro, 20-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harcard	|- ( card ` (har ` A ) ) = (har ` A )

Proof of Theorem harcard

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		harcl 8466	. 2 |- (har ` A ) e. On
2		harndom 8469	. . . . . . 7 |- -. (har ` A ) ~<_ A
3		simpll 790	. . . . . . . . 9 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> x e. On )
4		simpr 477	. . . . . . . . . . 11 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> y e. (har ` A ) )
5		elharval 8468	. . . . . . . . . . 11 |- ( y e. (har ` A ) <-> ( y e. On /\ y ~<_ A ) )
6	4, 5	sylib 208	. . . . . . . . . 10 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> ( y e. On /\ y ~<_ A ) )
7	6	simpld 475	. . . . . . . . 9 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> y e. On )
8		ontri1 5757	. . . . . . . . 9 |- ( ( x e. On /\ y e. On ) -> ( x C_ y <-> -. y e. x ) )
9	3, 7, 8	syl2anc 693	. . . . . . . 8 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> ( x C_ y <-> -. y e. x ) )
10		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) /\ x C_ y ) -> (har ` A ) ~~ x )
11		vex 3203	. . . . . . . . . . . 12 |- y e. _V
12		ssdomg 8001	. . . . . . . . . . . 12 |- ( y e. _V -> ( x C_ y -> x ~<_ y ) )
13	11, 12	ax-mp 5	. . . . . . . . . . 11 |- ( x C_ y -> x ~<_ y )
14	6	simprd 479	. . . . . . . . . . 11 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> y ~<_ A )
15		domtr 8009	. . . . . . . . . . 11 |- ( ( x ~<_ y /\ y ~<_ A ) -> x ~<_ A )
16	13, 14, 15	syl2anr 495	. . . . . . . . . 10 |- ( ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) /\ x C_ y ) -> x ~<_ A )
17		endomtr 8014	. . . . . . . . . 10 |- ( ( (har ` A ) ~~ x /\ x ~<_ A ) -> (har ` A ) ~<_ A )
18	10, 16, 17	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) /\ x C_ y ) -> (har ` A ) ~<_ A )
19	18	ex 450	. . . . . . . 8 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> ( x C_ y -> (har ` A ) ~<_ A ) )
20	9, 19	sylbird 250	. . . . . . 7 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> ( -. y e. x -> (har ` A ) ~<_ A ) )
21	2, 20	mt3i 141	. . . . . 6 |- ( ( ( x e. On /\ (har ` A ) ~~ x ) /\ y e. (har ` A ) ) -> y e. x )
22	21	ex 450	. . . . 5 |- ( ( x e. On /\ (har ` A ) ~~ x ) -> ( y e. (har ` A ) -> y e. x ) )
23	22	ssrdv 3609	. . . 4 |- ( ( x e. On /\ (har ` A ) ~~ x ) -> (har ` A ) C_ x )
24	23	ex 450	. . 3 |- ( x e. On -> ( (har ` A ) ~~ x -> (har ` A ) C_ x ) )
25	24	rgen 2922	. 2 |- A. x e. On ( (har ` A ) ~~ x -> (har ` A ) C_ x )
26		iscard2 8802	. 2 |- ( ( card ` (har ` A ) ) = (har ` A ) <-> ( (har ` A ) e. On /\ A. x e. On ( (har ` A ) ~~ x -> (har ` A ) C_ x ) ) )
27	1, 25, 26	mpbir2an 955	1 |- ( card ` (har ` A ) ) = (har ` A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Oncon0 5723   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953  harchar 8461   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-wrecs 7407  df-recs 7468  df-er 7742  df-en 7956  df-dom 7957  df-oi 8415  df-har 8463  df-card 8765

This theorem is referenced by:  cardprclem  8805  alephcard  8893  pwcfsdom  9405  hargch  9495