Theorem cardmin2 8824

Description: The smallest ordinal that strictly dominates a set is a cardinal, if it exists. (Contributed by Mario Carneiro, 2-Feb-2013.)

Assertion

Ref	Expression
cardmin2	|- ( E. x e. On A ~< x <-> ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } )

Distinct variable group:   x, A

Proof of Theorem cardmin2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onintrab2 7002	. . . 4 |- ( E. x e. On A ~< x <-> |^| { x e. On | A ~< x } e. On )
2	1	biimpi 206	. . 3 |- ( E. x e. On A ~< x -> |^| { x e. On | A ~< x } e. On )
3	2	adantr 481	. . . . . 6 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> |^| { x e. On | A ~< x } e. On )
4		eloni 5733	. . . . . . . 8 |- ( |^| { x e. On | A ~< x } e. On -> Ord |^| { x e. On | A ~< x } )
5		ordelss 5739	. . . . . . . 8 |- ( ( Ord |^| { x e. On | A ~< x } /\ y e. |^| { x e. On | A ~< x } ) -> y C_ |^| { x e. On | A ~< x } )
6	4, 5	sylan 488	. . . . . . 7 |- ( ( |^| { x e. On | A ~< x } e. On /\ y e. |^| { x e. On | A ~< x } ) -> y C_ |^| { x e. On | A ~< x } )
7	1, 6	sylanb 489	. . . . . 6 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> y C_ |^| { x e. On | A ~< x } )
8		ssdomg 8001	. . . . . 6 |- ( |^| { x e. On | A ~< x } e. On -> ( y C_ |^| { x e. On | A ~< x } -> y ~<_ |^| { x e. On | A ~< x } ) )
9	3, 7, 8	sylc 65	. . . . 5 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> y ~<_ |^| { x e. On | A ~< x } )
10		onelon 5748	. . . . . . . 8 |- ( ( |^| { x e. On | A ~< x } e. On /\ y e. |^| { x e. On | A ~< x } ) -> y e. On )
11	1, 10	sylanb 489	. . . . . . 7 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> y e. On )
12		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ x A
13		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ x ~<
14		nfrab1 3122	. . . . . . . . . . . . . . 15 |- F/_ x { x e. On | A ~< x }
15	14	nfint 4486	. . . . . . . . . . . . . 14 |- F/_ x |^| { x e. On | A ~< x }
16	12, 13, 15	nfbr 4699	. . . . . . . . . . . . 13 |- F/ x A ~< |^| { x e. On | A ~< x }
17		breq2 4657	. . . . . . . . . . . . 13 |- ( x = |^| { x e. On | A ~< x } -> ( A ~< x <-> A ~< |^| { x e. On | A ~< x } ) )
18	16, 17	onminsb 6999	. . . . . . . . . . . 12 |- ( E. x e. On A ~< x -> A ~< |^| { x e. On | A ~< x } )
19		sdomentr 8094	. . . . . . . . . . . 12 |- ( ( A ~< |^| { x e. On | A ~< x } /\ |^| { x e. On | A ~< x } ~~ y ) -> A ~< y )
20	18, 19	sylan 488	. . . . . . . . . . 11 |- ( ( E. x e. On A ~< x /\ |^| { x e. On | A ~< x } ~~ y ) -> A ~< y )
21		breq2 4657	. . . . . . . . . . . . . 14 |- ( x = y -> ( A ~< x <-> A ~< y ) )
22	21	elrab 3363	. . . . . . . . . . . . 13 |- ( y e. { x e. On | A ~< x } <-> ( y e. On /\ A ~< y ) )
23		ssrab2 3687	. . . . . . . . . . . . . 14 |- { x e. On | A ~< x } C_ On
24		onnmin 7003	. . . . . . . . . . . . . 14 |- ( ( { x e. On | A ~< x } C_ On /\ y e. { x e. On | A ~< x } ) -> -. y e. |^| { x e. On | A ~< x } )
25	23, 24	mpan 706	. . . . . . . . . . . . 13 |- ( y e. { x e. On | A ~< x } -> -. y e. |^| { x e. On | A ~< x } )
26	22, 25	sylbir 225	. . . . . . . . . . . 12 |- ( ( y e. On /\ A ~< y ) -> -. y e. |^| { x e. On | A ~< x } )
27	26	expcom 451	. . . . . . . . . . 11 |- ( A ~< y -> ( y e. On -> -. y e. |^| { x e. On | A ~< x } ) )
28	20, 27	syl 17	. . . . . . . . . 10 |- ( ( E. x e. On A ~< x /\ |^| { x e. On | A ~< x } ~~ y ) -> ( y e. On -> -. y e. |^| { x e. On | A ~< x } ) )
29	28	impancom 456	. . . . . . . . 9 |- ( ( E. x e. On A ~< x /\ y e. On ) -> ( |^| { x e. On | A ~< x } ~~ y -> -. y e. |^| { x e. On | A ~< x } ) )
30	29	con2d 129	. . . . . . . 8 |- ( ( E. x e. On A ~< x /\ y e. On ) -> ( y e. |^| { x e. On | A ~< x } -> -. |^| { x e. On | A ~< x } ~~ y ) )
31	30	impancom 456	. . . . . . 7 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> ( y e. On -> -. |^| { x e. On | A ~< x } ~~ y ) )
32	11, 31	mpd 15	. . . . . 6 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> -. |^| { x e. On | A ~< x } ~~ y )
33		ensym 8005	. . . . . 6 |- ( y ~~ |^| { x e. On | A ~< x } -> |^| { x e. On | A ~< x } ~~ y )
34	32, 33	nsyl 135	. . . . 5 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> -. y ~~ |^| { x e. On | A ~< x } )
35		brsdom 7978	. . . . 5 |- ( y ~< |^| { x e. On | A ~< x } <-> ( y ~<_ |^| { x e. On | A ~< x } /\ -. y ~~ |^| { x e. On | A ~< x } ) )
36	9, 34, 35	sylanbrc 698	. . . 4 |- ( ( E. x e. On A ~< x /\ y e. |^| { x e. On | A ~< x } ) -> y ~< |^| { x e. On | A ~< x } )
37	36	ralrimiva 2966	. . 3 |- ( E. x e. On A ~< x -> A. y e. |^| { x e. On | A ~< x } y ~< |^| { x e. On | A ~< x } )
38		iscard 8801	. . 3 |- ( ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } <-> ( |^| { x e. On | A ~< x } e. On /\ A. y e. |^| { x e. On | A ~< x } y ~< |^| { x e. On | A ~< x } ) )
39	2, 37, 38	sylanbrc 698	. 2 |- ( E. x e. On A ~< x -> ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } )
40		vprc 4796	. . . . . 6 |- -. _V e. _V
41		inteq 4478	. . . . . . . 8 |- ( { x e. On | A ~< x } = (/) -> |^| { x e. On | A ~< x } = |^| (/) )
42		int0 4490	. . . . . . . 8 |- |^| (/) = _V
43	41, 42	syl6eq 2672	. . . . . . 7 |- ( { x e. On | A ~< x } = (/) -> |^| { x e. On | A ~< x } = _V )
44	43	eleq1d 2686	. . . . . 6 |- ( { x e. On | A ~< x } = (/) -> ( |^| { x e. On | A ~< x } e. _V <-> _V e. _V ) )
45	40, 44	mtbiri 317	. . . . 5 |- ( { x e. On | A ~< x } = (/) -> -. |^| { x e. On | A ~< x } e. _V )
46		fvex 6201	. . . . . 6 |- ( card ` |^| { x e. On | A ~< x } ) e. _V
47		eleq1 2689	. . . . . 6 |- ( ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } -> ( ( card ` |^| { x e. On | A ~< x } ) e. _V <-> |^| { x e. On | A ~< x } e. _V ) )
48	46, 47	mpbii 223	. . . . 5 |- ( ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } -> |^| { x e. On | A ~< x } e. _V )
49	45, 48	nsyl 135	. . . 4 |- ( { x e. On | A ~< x } = (/) -> -. ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } )
50	49	necon2ai 2823	. . 3 |- ( ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } -> { x e. On | A ~< x } =/= (/) )
51		rabn0 3958	. . 3 |- ( { x e. On | A ~< x } =/= (/) <-> E. x e. On A ~< x )
52	50, 51	sylib 208	. 2 |- ( ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } -> E. x e. On A ~< x )
53	39, 52	impbii 199	1 |- ( E. x e. On A ~< x <-> ( card ` |^| { x e. On | A ~< x } ) = |^| { x e. On | A ~< x } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |^|cint 4475   class class class wbr 4653   Ord word 5722   Oncon0 5723   ` cfv 5888   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-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-ord 5726  df-on 5727  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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by: (None)