Theorem harval2 8823

Description: An alternate expression for the Hartogs number of a well-orderable set. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
harval2	|- ( A e. dom card -> (har ` A ) = |^| { x e. On | A ~< x } )

Distinct variable group:   x, A

Proof of Theorem harval2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		harval 8467	. . . . . . 7 |- ( A e. dom card -> (har ` A ) = { y e. On | y ~<_ A } )
2	1	adantr 481	. . . . . 6 |- ( ( A e. dom card /\ ( x e. On /\ A ~< x ) ) -> (har ` A ) = { y e. On | y ~<_ A } )
3		domsdomtr 8095	. . . . . . . . . . . . 13 |- ( ( y ~<_ A /\ A ~< x ) -> y ~< x )
4		sdomel 8107	. . . . . . . . . . . . 13 |- ( ( y e. On /\ x e. On ) -> ( y ~< x -> y e. x ) )
5	3, 4	syl5 34	. . . . . . . . . . . 12 |- ( ( y e. On /\ x e. On ) -> ( ( y ~<_ A /\ A ~< x ) -> y e. x ) )
6	5	imp 445	. . . . . . . . . . 11 |- ( ( ( y e. On /\ x e. On ) /\ ( y ~<_ A /\ A ~< x ) ) -> y e. x )
7	6	an4s 869	. . . . . . . . . 10 |- ( ( ( y e. On /\ y ~<_ A ) /\ ( x e. On /\ A ~< x ) ) -> y e. x )
8	7	ancoms 469	. . . . . . . . 9 |- ( ( ( x e. On /\ A ~< x ) /\ ( y e. On /\ y ~<_ A ) ) -> y e. x )
9	8	3impb 1260	. . . . . . . 8 |- ( ( ( x e. On /\ A ~< x ) /\ y e. On /\ y ~<_ A ) -> y e. x )
10	9	rabssdv 3682	. . . . . . 7 |- ( ( x e. On /\ A ~< x ) -> { y e. On | y ~<_ A } C_ x )
11	10	adantl 482	. . . . . 6 |- ( ( A e. dom card /\ ( x e. On /\ A ~< x ) ) -> { y e. On | y ~<_ A } C_ x )
12	2, 11	eqsstrd 3639	. . . . 5 |- ( ( A e. dom card /\ ( x e. On /\ A ~< x ) ) -> (har ` A ) C_ x )
13	12	expr 643	. . . 4 |- ( ( A e. dom card /\ x e. On ) -> ( A ~< x -> (har ` A ) C_ x ) )
14	13	ralrimiva 2966	. . 3 |- ( A e. dom card -> A. x e. On ( A ~< x -> (har ` A ) C_ x ) )
15		ssintrab 4500	. . 3 |- ( (har ` A ) C_ |^| { x e. On | A ~< x } <-> A. x e. On ( A ~< x -> (har ` A ) C_ x ) )
16	14, 15	sylibr 224	. 2 |- ( A e. dom card -> (har ` A ) C_ |^| { x e. On | A ~< x } )
17		harcl 8466	. . . . 5 |- (har ` A ) e. On
18	17	a1i 11	. . . 4 |- ( A e. dom card -> (har ` A ) e. On )
19		harsdom 8821	. . . 4 |- ( A e. dom card -> A ~< (har ` A ) )
20		breq2 4657	. . . . 5 |- ( x = (har ` A ) -> ( A ~< x <-> A ~< (har ` A ) ) )
21	20	elrab 3363	. . . 4 |- ( (har ` A ) e. { x e. On | A ~< x } <-> ( (har ` A ) e. On /\ A ~< (har ` A ) ) )
22	18, 19, 21	sylanbrc 698	. . 3 |- ( A e. dom card -> (har ` A ) e. { x e. On | A ~< x } )
23		intss1 4492	. . 3 |- ( (har ` A ) e. { x e. On | A ~< x } -> |^| { x e. On | A ~< x } C_ (har ` A ) )
24	22, 23	syl 17	. 2 |- ( A e. dom card -> |^| { x e. On | A ~< x } C_ (har ` A ) )
25	16, 24	eqssd 3620	1 |- ( A e. dom card -> (har ` A ) = |^| { x e. On | A ~< x } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   |^|cint 4475   class class class wbr 4653   dom cdm 5114   Oncon0 5723   ` cfv 5888   ~<_ cdom 7953   ~< csdm 7954  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-sdom 7958  df-oi 8415  df-har 8463  df-card 8765

This theorem is referenced by:  alephnbtwn  8894