Theorem card2inf 8073

Description: The definition cardval2 8427 has the curious property that for non-numerable sets (for which ndmfv 5902 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 4423	. . . . 5 |- ( x = (/) -> ( x ~< A <-> (/) ~< A ) )
2		breq1 4423	. . . . 5 |- ( x = n -> ( x ~< A <-> n ~< A ) )
3		breq1 4423	. . . . 5 |- ( x = suc n -> ( x ~< A <-> suc n ~< A ) )
4		0elon 5492	. . . . . . . 8 |- (/) e. On
5		breq1 4423	. . . . . . . . 9 |- ( y = (/) -> ( y ~~ A <-> (/) ~~ A ) )
6	5	rspcev 3182	. . . . . . . 8 |- ( ( (/) e. On /\ (/) ~~ A ) -> E. y e. On y ~~ A )
7	4, 6	mpan 674	. . . . . . 7 |- ( (/) ~~ A -> E. y e. On y ~~ A )
8	7	con3i 140	. . . . . 6 |- ( -. E. y e. On y ~~ A -> -. (/) ~~ A )
9		card2inf.1	. . . . . . . 8 |- A e. _V
10	9	0dom 7705	. . . . . . 7 |- (/) ~<_ A
11		brsdom 7596	. . . . . . 7 |- ( (/) ~< A <-> ( (/) ~<_ A /\ -. (/) ~~ A ) )
12	10, 11	mpbiran 926	. . . . . 6 |- ( (/) ~< A <-> -. (/) ~~ A )
13	8, 12	sylibr 215	. . . . 5 |- ( -. E. y e. On y ~~ A -> (/) ~< A )
14		sucdom2 7771	. . . . . . . 8 |- ( n ~< A -> suc n ~<_ A )
15	14	ad2antll 733	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~<_ A )
16		nnon 6709	. . . . . . . . . 10 |- ( n e. om -> n e. On )
17		suceloni 6651	. . . . . . . . . 10 |- ( n e. On -> suc n e. On )
18		breq1 4423	. . . . . . . . . . . 12 |- ( y = suc n -> ( y ~~ A <-> suc n ~~ A ) )
19	18	rspcev 3182	. . . . . . . . . . 11 |- ( ( suc n e. On /\ suc n ~~ A ) -> E. y e. On y ~~ A )
20	19	ex 435	. . . . . . . . . 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 442	. . . . . . . 8 |- ( ( n e. om /\ -. E. y e. On y ~~ A ) -> -. suc n ~~ A )
23	22	adantrr 721	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> -. suc n ~~ A )
24		brsdom 7596	. . . . . . 7 |- ( suc n ~< A <-> ( suc n ~<_ A /\ -. suc n ~~ A ) )
25	15, 23, 24	sylanbrc 668	. . . . . 6 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~< A )
26	25	exp32 608	. . . . 5 |- ( n e. om -> ( -. E. y e. On y ~~ A -> ( n ~< A -> suc n ~< A ) ) )
27	1, 2, 3, 13, 26	finds2 6732	. . . 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 2837	. 2 |- ( -. E. y e. On y ~~ A -> A. x e. om x ~< A )
30		omsson 6707	. . 3 |- om C_ On
31		ssrab 3539	. . 3 |- ( om C_ { x e. On | x ~< A } <-> ( om C_ On /\ A. x e. om x ~< A ) )
32	30, 31	mpbiran 926	. 2 |- ( om C_ { x e. On | x ~< A } <-> A. x e. om x ~< A )
33	29, 32	sylibr 215	1 |- ( -. E. y e. On y ~~ A -> om C_ { x e. On | x ~< A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   e. wcel 1868   A.wral 2775   E.wrex 2776   {crab 2779   _Vcvv 3081   C_ wss 3436   (/)c0 3761   class class class wbr 4420   Oncon0 5439   suc csuc 5441   omcom 6703   ~~ cen 7571   ~<_ cdom 7572   ~< csdm 7573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-om 6704  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577

This theorem is referenced by: (None)