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Theorem elhf2 30990
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 30989 . 2  |-  ( A  e. Hf 
<->  E. x  e.  om  A  e.  ( R1 `  x ) )
2 omon 6729 . . 3  |-  ( om  e.  On  \/  om  =  On )
3 nnon 6724 . . . . . . . . 9  |-  ( x  e.  om  ->  x  e.  On )
4 elhf2.1 . . . . . . . . . 10  |-  A  e. 
_V
54rankr1a 8332 . . . . . . . . 9  |-  ( x  e.  On  ->  ( A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  x
) )
63, 5syl 17 . . . . . . . 8  |-  ( x  e.  om  ->  ( A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  x
) )
76adantl 472 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  x ) )
8 elnn 6728 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  x  /\  x  e.  om )  ->  ( rank `  A
)  e.  om )
98expcom 441 . . . . . . . 8  |-  ( x  e.  om  ->  (
( rank `  A )  e.  x  ->  ( rank `  A )  e.  om ) )
109adantl 472 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( ( rank `  A
)  e.  x  -> 
( rank `  A )  e.  om ) )
117, 10sylbid 223 . . . . . 6  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  ->  ( rank `  A
)  e.  om )
)
1211rexlimdva 2890 . . . . 5  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  ->  ( rank `  A )  e. 
om ) )
13 peano2 6739 . . . . . . . 8  |-  ( (
rank `  A )  e.  om  ->  suc  ( rank `  A )  e.  om )
1413adantr 471 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  suc  ( rank `  A
)  e.  om )
15 r1rankid 8355 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( R1 `  ( rank `  A ) ) )
164, 15mp1i 13 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
174elpw 3968 . . . . . . . . 9  |-  ( A  e.  ~P ( R1
`  ( rank `  A
) )  <->  A  C_  ( R1 `  ( rank `  A
) ) )
1816, 17sylibr 217 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ~P ( R1 `  ( rank `  A
) ) )
19 nnon 6724 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  om  ->  ( rank `  A )  e.  On )
20 r1suc 8266 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  On  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
2119, 20syl 17 . . . . . . . . 9  |-  ( (
rank `  A )  e.  om  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
2221adantr 471 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ~P ( R1 `  ( rank `  A
) ) )
2318, 22eleqtrrd 2542 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
24 fveq2 5887 . . . . . . . . 9  |-  ( x  =  suc  ( rank `  A )  ->  ( R1 `  x )  =  ( R1 `  suc  ( rank `  A )
) )
2524eleq2d 2524 . . . . . . . 8  |-  ( x  =  suc  ( rank `  A )  ->  ( A  e.  ( R1 `  x )  <->  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
2625rspcev 3161 . . . . . . 7  |-  ( ( suc  ( rank `  A
)  e.  om  /\  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2714, 23, 26syl2anc 671 . . . . . 6  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2827expcom 441 . . . . 5  |-  ( om  e.  On  ->  (
( rank `  A )  e.  om  ->  E. x  e.  om  A  e.  ( R1 `  x ) ) )
2912, 28impbid 195 . . . 4  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
304tz9.13 8287 . . . . . 6  |-  E. x  e.  On  A  e.  ( R1 `  x )
31 rankon 8291 . . . . . 6  |-  ( rank `  A )  e.  On
3230, 312th 247 . . . . 5  |-  ( E. x  e.  On  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  On )
33 rexeq 2999 . . . . . 6  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
34 eleq2 2528 . . . . . 6  |-  ( om  =  On  ->  (
( rank `  A )  e.  om  <->  ( rank `  A
)  e.  On ) )
3533, 34bibi12d 327 . . . . 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 241 . . . 4  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
3729, 36jaoi 385 . . 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 257 1  |-  ( A  e. Hf 
<->  ( rank `  A
)  e.  om )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    /\ wa 375    = wceq 1454    e. wcel 1897   E.wrex 2749   _Vcvv 3056    C_ wss 3415   ~Pcpw 3962   Oncon0 5441   suc csuc 5443   ` cfv 5600   omcom 6718   R1cr1 8258   rankcrnk 8259   Hf chf 30987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-reg 8132  ax-inf2 8171
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-r1 8260  df-rank 8261  df-hf 30988
This theorem is referenced by:  elhf2g  30991  hfsn  30994
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