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Theorem elhf2 29994
Description: Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
Hypothesis
Ref Expression
elhf2.1  |-  A  e. 
_V
Assertion
Ref Expression
elhf2  |-  ( A  e. Hf 
<->  ( rank `  A
)  e.  om )

Proof of Theorem elhf2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elhf 29993 . 2  |-  ( A  e. Hf 
<->  E. x  e.  om  A  e.  ( R1 `  x ) )
2 omon 6710 . . 3  |-  ( om  e.  On  \/  om  =  On )
3 nnon 6705 . . . . . . . . 9  |-  ( x  e.  om  ->  x  e.  On )
4 elhf2.1 . . . . . . . . . 10  |-  A  e. 
_V
54rankr1a 8271 . . . . . . . . 9  |-  ( x  e.  On  ->  ( A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  x
) )
63, 5syl 16 . . . . . . . 8  |-  ( x  e.  om  ->  ( A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  x
) )
76adantl 466 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  x ) )
8 elnn 6709 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  x  /\  x  e.  om )  ->  ( rank `  A
)  e.  om )
98expcom 435 . . . . . . . 8  |-  ( x  e.  om  ->  (
( rank `  A )  e.  x  ->  ( rank `  A )  e.  om ) )
109adantl 466 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( ( rank `  A
)  e.  x  -> 
( rank `  A )  e.  om ) )
117, 10sylbid 215 . . . . . 6  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  ->  ( rank `  A
)  e.  om )
)
1211rexlimdva 2949 . . . . 5  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  ->  ( rank `  A )  e. 
om ) )
13 peano2 6719 . . . . . . . 8  |-  ( (
rank `  A )  e.  om  ->  suc  ( rank `  A )  e.  om )
1413adantr 465 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  suc  ( rank `  A
)  e.  om )
15 r1rankid 8294 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( R1 `  ( rank `  A ) ) )
164, 15mp1i 12 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
174elpw 4021 . . . . . . . . 9  |-  ( A  e.  ~P ( R1
`  ( rank `  A
) )  <->  A  C_  ( R1 `  ( rank `  A
) ) )
1816, 17sylibr 212 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ~P ( R1 `  ( rank `  A
) ) )
19 nnon 6705 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  om  ->  ( rank `  A )  e.  On )
20 r1suc 8205 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  On  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
2119, 20syl 16 . . . . . . . . 9  |-  ( (
rank `  A )  e.  om  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
2221adantr 465 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ~P ( R1 `  ( rank `  A
) ) )
2318, 22eleqtrrd 2548 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
24 fveq2 5872 . . . . . . . . 9  |-  ( x  =  suc  ( rank `  A )  ->  ( R1 `  x )  =  ( R1 `  suc  ( rank `  A )
) )
2524eleq2d 2527 . . . . . . . 8  |-  ( x  =  suc  ( rank `  A )  ->  ( A  e.  ( R1 `  x )  <->  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
2625rspcev 3210 . . . . . . 7  |-  ( ( suc  ( rank `  A
)  e.  om  /\  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2714, 23, 26syl2anc 661 . . . . . 6  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2827expcom 435 . . . . 5  |-  ( om  e.  On  ->  (
( rank `  A )  e.  om  ->  E. x  e.  om  A  e.  ( R1 `  x ) ) )
2912, 28impbid 191 . . . 4  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
304tz9.13 8226 . . . . . 6  |-  E. x  e.  On  A  e.  ( R1 `  x )
31 rankon 8230 . . . . . 6  |-  ( rank `  A )  e.  On
3230, 312th 239 . . . . 5  |-  ( E. x  e.  On  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  On )
33 rexeq 3055 . . . . . 6  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
34 eleq2 2530 . . . . . 6  |-  ( om  =  On  ->  (
( rank `  A )  e.  om  <->  ( rank `  A
)  e.  On ) )
3533, 34bibi12d 321 . . . . 5  |-  ( om  =  On  ->  (
( E. x  e. 
om  A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  om )  <->  ( E. x  e.  On  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  On ) ) )
3632, 35mpbiri 233 . . . 4  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
3729, 36jaoi 379 . . 3  |-  ( ( om  e.  On  \/  om  =  On )  -> 
( E. x  e. 
om  A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  om ) )
382, 37ax-mp 5 . 2  |-  ( E. x  e.  om  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  om )
391, 38bitri 249 1  |-  ( A  e. Hf 
<->  ( rank `  A
)  e.  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   Oncon0 4887   suc csuc 4889   ` cfv 5594   omcom 6699   R1cr1 8197   rankcrnk 8198   Hf chf 29991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-r1 8199  df-rank 8200  df-hf 29992
This theorem is referenced by:  elhf2g  29995  hfsn  29998
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