Theorem card2inf 7874

Description: The definition cardval2 8265 has the curious property that for non-numerable sets (for which ndmfv 5816 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 4396	. . . . 5 |- ( x = (/) -> ( x ~< A <-> (/) ~< A ) )
2		breq1 4396	. . . . 5 |- ( x = n -> ( x ~< A <-> n ~< A ) )
3		breq1 4396	. . . . 5 |- ( x = suc n -> ( x ~< A <-> suc n ~< A ) )
4		0elon 4873	. . . . . . . 8 |- (/) e. On
5		breq1 4396	. . . . . . . . 9 |- ( y = (/) -> ( y ~~ A <-> (/) ~~ A ) )
6	5	rspcev 3172	. . . . . . . 8 |- ( ( (/) e. On /\ (/) ~~ A ) -> E. y e. On y ~~ A )
7	4, 6	mpan 670	. . . . . . 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 7544	. . . . . . 7 |- (/) ~<_ A
11		brsdom 7435	. . . . . . 7 |- ( (/) ~< A <-> ( (/) ~<_ A /\ -. (/) ~~ A ) )
12	10, 11	mpbiran 909	. . . . . 6 |- ( (/) ~< A <-> -. (/) ~~ A )
13	8, 12	sylibr 212	. . . . 5 |- ( -. E. y e. On y ~~ A -> (/) ~< A )
14		sucdom2 7611	. . . . . . . 8 |- ( n ~< A -> suc n ~<_ A )
15	14	ad2antll 728	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~<_ A )
16		nnon 6585	. . . . . . . . . 10 |- ( n e. om -> n e. On )
17		suceloni 6527	. . . . . . . . . 10 |- ( n e. On -> suc n e. On )
18		breq1 4396	. . . . . . . . . . . 12 |- ( y = suc n -> ( y ~~ A <-> suc n ~~ A ) )
19	18	rspcev 3172	. . . . . . . . . . 11 |- ( ( suc n e. On /\ suc n ~~ A ) -> E. y e. On y ~~ A )
20	19	ex 434	. . . . . . . . . 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 441	. . . . . . . 8 |- ( ( n e. om /\ -. E. y e. On y ~~ A ) -> -. suc n ~~ A )
23	22	adantrr 716	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> -. suc n ~~ A )
24		brsdom 7435	. . . . . . 7 |- ( suc n ~< A <-> ( suc n ~<_ A /\ -. suc n ~~ A ) )
25	15, 23, 24	sylanbrc 664	. . . . . 6 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~< A )
26	25	exp32 605	. . . . 5 |- ( n e. om -> ( -. E. y e. On y ~~ A -> ( n ~< A -> suc n ~< A ) ) )
27	1, 2, 3, 13, 26	finds2 6607	. . . 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 2823	. 2 |- ( -. E. y e. On y ~~ A -> A. x e. om x ~< A )
30		omsson 6583	. . 3 |- om C_ On
31		ssrab 3531	. . 3 |- ( om C_ { x e. On | x ~< A } <-> ( om C_ On /\ A. x e. om x ~< A ) )
32	30, 31	mpbiran 909	. 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 369   e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3071   C_ wss 3429   (/)c0 3738   class class class wbr 4393   Oncon0 4820   suc csuc 4822   omcom 6579   ~~ cen 7410   ~<_ cdom 7411   ~< csdm 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-om 6580  df-1o 7023  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416

This theorem is referenced by: (None)