Theorem card2inf 8075

Description: The definition cardval2 8430 has the curious property that for non-numerable sets (for which ndmfv 5894 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 4408	. . . . 5 |- ( x = (/) -> ( x ~< A <-> (/) ~< A ) )
2		breq1 4408	. . . . 5 |- ( x = n -> ( x ~< A <-> n ~< A ) )
3		breq1 4408	. . . . 5 |- ( x = suc n -> ( x ~< A <-> suc n ~< A ) )
4		0elon 5479	. . . . . . . 8 |- (/) e. On
5		breq1 4408	. . . . . . . . 9 |- ( y = (/) -> ( y ~~ A <-> (/) ~~ A ) )
6	5	rspcev 3152	. . . . . . . 8 |- ( ( (/) e. On /\ (/) ~~ A ) -> E. y e. On y ~~ A )
7	4, 6	mpan 677	. . . . . . 7 |- ( (/) ~~ A -> E. y e. On y ~~ A )
8	7	con3i 141	. . . . . 6 |- ( -. E. y e. On y ~~ A -> -. (/) ~~ A )
9		card2inf.1	. . . . . . . 8 |- A e. _V
10	9	0dom 7707	. . . . . . 7 |- (/) ~<_ A
11		brsdom 7597	. . . . . . 7 |- ( (/) ~< A <-> ( (/) ~<_ A /\ -. (/) ~~ A ) )
12	10, 11	mpbiran 930	. . . . . 6 |- ( (/) ~< A <-> -. (/) ~~ A )
13	8, 12	sylibr 216	. . . . 5 |- ( -. E. y e. On y ~~ A -> (/) ~< A )
14		sucdom2 7773	. . . . . . . 8 |- ( n ~< A -> suc n ~<_ A )
15	14	ad2antll 736	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~<_ A )
16		nnon 6703	. . . . . . . . . 10 |- ( n e. om -> n e. On )
17		suceloni 6645	. . . . . . . . . 10 |- ( n e. On -> suc n e. On )
18		breq1 4408	. . . . . . . . . . . 12 |- ( y = suc n -> ( y ~~ A <-> suc n ~~ A ) )
19	18	rspcev 3152	. . . . . . . . . . 11 |- ( ( suc n e. On /\ suc n ~~ A ) -> E. y e. On y ~~ A )
20	19	ex 436	. . . . . . . . . 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 443	. . . . . . . 8 |- ( ( n e. om /\ -. E. y e. On y ~~ A ) -> -. suc n ~~ A )
23	22	adantrr 724	. . . . . . 7 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> -. suc n ~~ A )
24		brsdom 7597	. . . . . . 7 |- ( suc n ~< A <-> ( suc n ~<_ A /\ -. suc n ~~ A ) )
25	15, 23, 24	sylanbrc 671	. . . . . 6 |- ( ( n e. om /\ ( -. E. y e. On y ~~ A /\ n ~< A ) ) -> suc n ~< A )
26	25	exp32 610	. . . . 5 |- ( n e. om -> ( -. E. y e. On y ~~ A -> ( n ~< A -> suc n ~< A ) ) )
27	1, 2, 3, 13, 26	finds2 6726	. . . 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 2802	. 2 |- ( -. E. y e. On y ~~ A -> A. x e. om x ~< A )
30		omsson 6701	. . 3 |- om C_ On
31		ssrab 3509	. . 3 |- ( om C_ { x e. On | x ~< A } <-> ( om C_ On /\ A. x e. om x ~< A ) )
32	30, 31	mpbiran 930	. 2 |- ( om C_ { x e. On | x ~< A } <-> A. x e. om x ~< A )
33	29, 32	sylibr 216	1 |- ( -. E. y e. On y ~~ A -> om C_ { x e. On | x ~< A } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   e. wcel 1889   A.wral 2739   E.wrex 2740   {crab 2743   _Vcvv 3047   C_ wss 3406   (/)c0 3733   class class class wbr 4405   Oncon0 5426   suc csuc 5428   omcom 6697   ~~ 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 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-om 6698  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577

This theorem is referenced by: (None)