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Theorem elhf2 26116
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 26115 . 2  |-  ( A  e. Hf 
<->  E. x  e.  om  A  e.  ( R1 `  x ) )
2 omon 4856 . . 3  |-  ( om  e.  On  \/  om  =  On )
3 nnon 4851 . . . . . . . . 9  |-  ( x  e.  om  ->  x  e.  On )
4 elhf2.1 . . . . . . . . . 10  |-  A  e. 
_V
54rankr1a 7762 . . . . . . . . 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 453 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  x ) )
8 elnn 4855 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  x  /\  x  e.  om )  ->  ( rank `  A
)  e.  om )
98expcom 425 . . . . . . . 8  |-  ( x  e.  om  ->  (
( rank `  A )  e.  x  ->  ( rank `  A )  e.  om ) )
109adantl 453 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( ( rank `  A
)  e.  x  -> 
( rank `  A )  e.  om ) )
117, 10sylbid 207 . . . . . 6  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  ->  ( rank `  A
)  e.  om )
)
1211rexlimdva 2830 . . . . 5  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  ->  ( rank `  A )  e. 
om ) )
13 peano2 4865 . . . . . . . 8  |-  ( (
rank `  A )  e.  om  ->  suc  ( rank `  A )  e.  om )
1413adantr 452 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  suc  ( rank `  A
)  e.  om )
15 r1rankid 7785 . . . . . . . . . 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 3805 . . . . . . . . 9  |-  ( A  e.  ~P ( R1
`  ( rank `  A
) )  <->  A  C_  ( R1 `  ( rank `  A
) ) )
1816, 17sylibr 204 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ~P ( R1 `  ( rank `  A
) ) )
19 nnon 4851 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  om  ->  ( rank `  A )  e.  On )
20 r1suc 7696 . . . . . . . . . 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 452 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ~P ( R1 `  ( rank `  A
) ) )
2318, 22eleqtrrd 2513 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
24 fveq2 5728 . . . . . . . . 9  |-  ( x  =  suc  ( rank `  A )  ->  ( R1 `  x )  =  ( R1 `  suc  ( rank `  A )
) )
2524eleq2d 2503 . . . . . . . 8  |-  ( x  =  suc  ( rank `  A )  ->  ( A  e.  ( R1 `  x )  <->  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
2625rspcev 3052 . . . . . . 7  |-  ( ( suc  ( rank `  A
)  e.  om  /\  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2714, 23, 26syl2anc 643 . . . . . 6  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2827expcom 425 . . . . 5  |-  ( om  e.  On  ->  (
( rank `  A )  e.  om  ->  E. x  e.  om  A  e.  ( R1 `  x ) ) )
2912, 28impbid 184 . . . 4  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
304tz9.13 7717 . . . . . 6  |-  E. x  e.  On  A  e.  ( R1 `  x )
31 rankon 7721 . . . . . 6  |-  ( rank `  A )  e.  On
3230, 312th 231 . . . . 5  |-  ( E. x  e.  On  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  On )
33 rexeq 2905 . . . . . 6  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
34 eleq2 2497 . . . . . 6  |-  ( om  =  On  ->  (
( rank `  A )  e.  om  <->  ( rank `  A
)  e.  On ) )
3533, 34bibi12d 313 . . . . 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 225 . . . 4  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
3729, 36jaoi 369 . . 3  |-  ( ( om  e.  On  \/  om  =  On )  -> 
( E. x  e. 
om  A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  om ) )
382, 37ax-mp 8 . 2  |-  ( E. x  e.  om  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  om )
391, 38bitri 241 1  |-  ( A  e. Hf 
<->  ( rank `  A
)  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   Oncon0 4581   suc csuc 4583   omcom 4845   ` cfv 5454   R1cr1 7688   rankcrnk 7689   Hf chf 26113
This theorem is referenced by:  elhf2g  26117  hfsn  26120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-r1 7690  df-rank 7691  df-hf 26114
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