Theorem card2inf 7973

Description: The definition cardval2 8363 has the curious property that for non-numerable sets (for which ndmfv 5872 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 4442	. . . . 5 |- ( x = (/) -> ( x ~< A <-> (/) ~< A ) )
2		breq1 4442	. . . . 5 |- ( x = n -> ( x ~< A <-> n ~< A ) )
3		breq1 4442	. . . . 5 |- ( x = suc n -> ( x ~< A <-> suc n ~< A ) )
4		0elon 4920	. . . . . . . 8 |- (/) e. On
5		breq1 4442	. . . . . . . . 9 |- ( y = (/) -> ( y ~~ A <-> (/) ~~ A ) )
6	5	rspcev 3207	. . . . . . . 8 |- ( ( (/) e. On /\ (/) ~~ A ) -> E. y e. On y ~~ A )
7	4, 6	mpan 668	. . . . . . 7 |- ( (/) ~~ A -> E. y e. On y ~~ A )
8	7	con3i 135	. . . . . 6 |- ( -. E. y e. On y ~~ A -> -. (/) ~~ A )
9		card2inf.1	. . . . . . . 8 |- A e. _V
10	9	0dom 7640	. . . . . . 7 |- (/) ~<_ A
11		brsdom 7531	. . . . . . 7 |- ( (/) ~< A <-> ( (/) ~<_ A /\ -. (/) ~~ A ) )
12	10, 11	mpbiran 916	. . . . . 6 |- ( (/) ~< A <-> -. (/) ~~ A )
13	8, 12	sylibr 212	. . . . 5 |- ( -. E. y e. On y ~~ A -> (/) ~< A )
14		sucdom2 7707	. . . . . . . 8 |- ( n ~< A -> suc n ~<_ A )
15	14	ad2antll 726	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~<_ A )
16		nnon 6679	. . . . . . . . . 10 |- ( n e. om -> n e. On )
17		suceloni 6621	. . . . . . . . . 10 |- ( n e. On -> suc n e. On )
18		breq1 4442	. . . . . . . . . . . 12 |- ( y = suc n -> ( y ~~ A <-> suc n ~~ A ) )
19	18	rspcev 3207	. . . . . . . . . . 11 |- ( ( suc n e. On /\ suc n ~~ A ) -> E. y e. On y ~~ A )
20	19	ex 432	. . . . . . . . . 10 |- ( suc n e. On -> ( suc n ~~ A -> E. y e. On y ~~ A ) )
21	16, 17, 20	3syl 20	. . . . . . . . 9 |- ( n e. om -> ( suc n ~~ A -> E. y e. On y ~~ A ) )
22	21	con3dimp 439	. . . . . . . 8 |- ( ( n e. om /\ -. E. y e. On y ~~ A ) -> -. suc n ~~ A )
23	22	adantrr 714	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> -. suc n ~~ A )
24		brsdom 7531	. . . . . . 7 |- ( suc n ~< A <-> ( suc n ~<_ A /\ -. suc n ~~ A ) )
25	15, 23, 24	sylanbrc 662	. . . . . 6 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~< A )
26	25	exp32 603	. . . . 5 |- ( n e. om -> ( -. E. y e. On y ~~ A -> ( n ~< A -> suc n ~< A ) ) )
27	1, 2, 3, 13, 26	finds2 6701	. . . 4 |- ( x e. om -> ( -. E. y e. On y ~~ A -> x ~< A ) )
28	27	com12 31	. . 3 |- ( -. E. y e. On y ~~ A -> ( x e. om -> x ~< A ) )
29	28	ralrimiv 2866	. 2 |- ( -. E. y e. On y ~~ A -> A. x e. om x ~< A )
30		omsson 6677	. . 3 |- om C_ On
31		ssrab 3564	. . 3 |- ( om C_ { x e. On | x ~< A } <-> ( om C_ On /\ A. x e. om x ~< A ) )
32	30, 31	mpbiran 916	. 2 |- ( om C_ { x e. On | x ~< A } <-> A. x e. om x ~< A )
33	29, 32	sylibr 212	1 |- ( -. E. y e. On y ~~ A -> om C_ { x e. On | x ~< A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   C_ wss 3461   (/)c0 3783   class class class wbr 4439   Oncon0 4867   suc csuc 4869   omcom 6673   ~~ cen 7506   ~<_ cdom 7507   ~< csdm 7508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-om 6674  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512

This theorem is referenced by: (None)