Theorem card2inf 8460

Description: The definition cardval2 8817 has the curious property that for non-numerable sets (for which ndmfv 6218 yields (/)), it still evaluates to a nonempty set, and indeed it contains om. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 27-Apr-2015.)

Hypothesis

Ref	Expression
card2inf.1	|- A e. _V

Assertion

Ref	Expression
card2inf	|- ( -. E. y e. On y ~~ A -> om C_ { x e. On | x ~< A } )

Distinct variable group:   x, A, y

Proof of Theorem card2inf

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . 5 |- ( x = (/) -> ( x ~< A <-> (/) ~< A ) )
2		breq1 4656	. . . . 5 |- ( x = n -> ( x ~< A <-> n ~< A ) )
3		breq1 4656	. . . . 5 |- ( x = suc n -> ( x ~< A <-> suc n ~< A ) )
4		0elon 5778	. . . . . . . 8 |- (/) e. On
5		breq1 4656	. . . . . . . . 9 |- ( y = (/) -> ( y ~~ A <-> (/) ~~ A ) )
6	5	rspcev 3309	. . . . . . . 8 |- ( ( (/) e. On /\ (/) ~~ A ) -> E. y e. On y ~~ A )
7	4, 6	mpan 706	. . . . . . 7 |- ( (/) ~~ A -> E. y e. On y ~~ A )
8	7	con3i 150	. . . . . 6 |- ( -. E. y e. On y ~~ A -> -. (/) ~~ A )
9		card2inf.1	. . . . . . . 8 |- A e. _V
10	9	0dom 8090	. . . . . . 7 |- (/) ~<_ A
11		brsdom 7978	. . . . . . 7 |- ( (/) ~< A <-> ( (/) ~<_ A /\ -. (/) ~~ A ) )
12	10, 11	mpbiran 953	. . . . . 6 |- ( (/) ~< A <-> -. (/) ~~ A )
13	8, 12	sylibr 224	. . . . 5 |- ( -. E. y e. On y ~~ A -> (/) ~< A )
14		sucdom2 8156	. . . . . . . 8 |- ( n ~< A -> suc n ~<_ A )
15	14	ad2antll 765	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~<_ A )
16		nnon 7071	. . . . . . . . . 10 |- ( n e. om -> n e. On )
17		suceloni 7013	. . . . . . . . . 10 |- ( n e. On -> suc n e. On )
18		breq1 4656	. . . . . . . . . . . 12 |- ( y = suc n -> ( y ~~ A <-> suc n ~~ A ) )
19	18	rspcev 3309	. . . . . . . . . . 11 |- ( ( suc n e. On /\ suc n ~~ A ) -> E. y e. On y ~~ A )
20	19	ex 450	. . . . . . . . . 10 |- ( suc n e. On -> ( suc n ~~ A -> E. y e. On y ~~ A ) )
21	16, 17, 20	3syl 18	. . . . . . . . 9 |- ( n e. om -> ( suc n ~~ A -> E. y e. On y ~~ A ) )
22	21	con3dimp 457	. . . . . . . 8 |- ( ( n e. om /\ -. E. y e. On y ~~ A ) -> -. suc n ~~ A )
23	22	adantrr 753	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> -. suc n ~~ A )
24		brsdom 7978	. . . . . . 7 |- ( suc n ~< A <-> ( suc n ~<_ A /\ -. suc n ~~ A ) )
25	15, 23, 24	sylanbrc 698	. . . . . 6 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~< A )
26	25	exp32 631	. . . . 5 |- ( n e. om -> ( -. E. y e. On y ~~ A -> ( n ~< A -> suc n ~< A ) ) )
27	1, 2, 3, 13, 26	finds2 7094	. . . 4 |- ( x e. om -> ( -. E. y e. On y ~~ A -> x ~< A ) )
28	27	com12 32	. . 3 |- ( -. E. y e. On y ~~ A -> ( x e. om -> x ~< A ) )
29	28	ralrimiv 2965	. 2 |- ( -. E. y e. On y ~~ A -> A. x e. om x ~< A )
30		omsson 7069	. . 3 |- om C_ On
31		ssrab 3680	. . 3 |- ( om C_ { x e. On | x ~< A } <-> ( om C_ On /\ A. x e. om x ~< A ) )
32	30, 31	mpbiran 953	. 2 |- ( om C_ { x e. On | x ~< A } <-> A. x e. om x ~< A )
33	29, 32	sylibr 224	1 |- ( -. E. y e. On y ~~ A -> om C_ { x e. On | x ~< A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Oncon0 5723   suc csuc 5725   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954

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-br 4654  df-opab 4713  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-lim 5728  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by: (None)