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Theorem elhf2 24805
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 24804 . 2  |-  ( A  e. Hf 
<->  E. x  e.  om  A  e.  ( R1 `  x ) )
2 omon 4667 . . 3  |-  ( om  e.  On  \/  om  =  On )
3 nnon 4662 . . . . . . . . 9  |-  ( x  e.  om  ->  x  e.  On )
4 elhf2.1 . . . . . . . . . 10  |-  A  e. 
_V
54rankr1a 7508 . . . . . . . . 9  |-  ( x  e.  On  ->  ( A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  x
) )
63, 5syl 15 . . . . . . . 8  |-  ( x  e.  om  ->  ( A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  x
) )
76adantl 452 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  x ) )
8 elnn 4666 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  x  /\  x  e.  om )  ->  ( rank `  A
)  e.  om )
98expcom 424 . . . . . . . 8  |-  ( x  e.  om  ->  (
( rank `  A )  e.  x  ->  ( rank `  A )  e.  om ) )
109adantl 452 . . . . . . 7  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( ( rank `  A
)  e.  x  -> 
( rank `  A )  e.  om ) )
117, 10sylbid 206 . . . . . 6  |-  ( ( om  e.  On  /\  x  e.  om )  ->  ( A  e.  ( R1 `  x )  ->  ( rank `  A
)  e.  om )
)
1211rexlimdva 2667 . . . . 5  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  ->  ( rank `  A )  e. 
om ) )
13 peano2 4676 . . . . . . . 8  |-  ( (
rank `  A )  e.  om  ->  suc  ( rank `  A )  e.  om )
1413adantr 451 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  suc  ( rank `  A
)  e.  om )
15 r1rankid 7531 . . . . . . . . . 10  |-  ( A  e.  _V  ->  A  C_  ( R1 `  ( rank `  A ) ) )
164, 15mp1i 11 . . . . . . . . 9  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  C_  ( R1 `  ( rank `  A )
) )
174elpw 3631 . . . . . . . . 9  |-  ( A  e.  ~P ( R1
`  ( rank `  A
) )  <->  A  C_  ( R1 `  ( rank `  A
) ) )
1816, 17sylibr 203 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ~P ( R1 `  ( rank `  A
) ) )
19 nnon 4662 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  om  ->  ( rank `  A )  e.  On )
20 r1suc 7442 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  On  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
2119, 20syl 15 . . . . . . . . 9  |-  ( (
rank `  A )  e.  om  ->  ( R1 ` 
suc  ( rank `  A
) )  =  ~P ( R1 `  ( rank `  A ) ) )
2221adantr 451 . . . . . . . 8  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  -> 
( R1 `  suc  ( rank `  A )
)  =  ~P ( R1 `  ( rank `  A
) ) )
2318, 22eleqtrrd 2360 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
24 fveq2 5525 . . . . . . . . 9  |-  ( x  =  suc  ( rank `  A )  ->  ( R1 `  x )  =  ( R1 `  suc  ( rank `  A )
) )
2524eleq2d 2350 . . . . . . . 8  |-  ( x  =  suc  ( rank `  A )  ->  ( A  e.  ( R1 `  x )  <->  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
2625rspcev 2884 . . . . . . 7  |-  ( ( suc  ( rank `  A
)  e.  om  /\  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2714, 23, 26syl2anc 642 . . . . . 6  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  E. x  e.  om  A  e.  ( R1 `  x ) )
2827expcom 424 . . . . 5  |-  ( om  e.  On  ->  (
( rank `  A )  e.  om  ->  E. x  e.  om  A  e.  ( R1 `  x ) ) )
2912, 28impbid 183 . . . 4  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
30 tz9.13g 7464 . . . . . . 7  |-  ( A  e.  _V  ->  E. x  e.  On  A  e.  ( R1 `  x ) )
314, 30ax-mp 8 . . . . . 6  |-  E. x  e.  On  A  e.  ( R1 `  x )
32 rankon 7467 . . . . . 6  |-  ( rank `  A )  e.  On
3331, 322th 230 . . . . 5  |-  ( E. x  e.  On  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  On )
34 rexeq 2737 . . . . . 6  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
35 eleq2 2344 . . . . . 6  |-  ( om  =  On  ->  (
( rank `  A )  e.  om  <->  ( rank `  A
)  e.  On ) )
3634, 35bibi12d 312 . . . . 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 ) ) )
3733, 36mpbiri 224 . . . 4  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  ( rank `  A )  e.  om ) )
3829, 37jaoi 368 . . 3  |-  ( ( om  e.  On  \/  om  =  On )  -> 
( E. x  e. 
om  A  e.  ( R1 `  x )  <-> 
( rank `  A )  e.  om ) )
392, 38ax-mp 8 . 2  |-  ( E. x  e.  om  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  om )
401, 39bitri 240 1  |-  ( A  e. Hf 
<->  ( rank `  A
)  e.  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   Oncon0 4392   suc csuc 4394   omcom 4656   ` cfv 5255   R1cr1 7434   rankcrnk 7435   Hf chf 24802
This theorem is referenced by:  elhf2g  24806  hfsn  24809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-r1 7436  df-rank 7437  df-hf 24803
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