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Theorem elhf2 24877
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 24876 . 2  |-  ( A  e. Hf 
<->  E. x  e.  om  A  e.  ( R1 `  x ) )
2 omon 4683 . . 3  |-  ( om  e.  On  \/  om  =  On )
3 nnon 4678 . . . . . . . . 9  |-  ( x  e.  om  ->  x  e.  On )
4 elhf2.1 . . . . . . . . . 10  |-  A  e. 
_V
54rankr1a 7524 . . . . . . . . 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 4682 . . . . . . . . 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 2680 . . . . 5  |-  ( om  e.  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  ->  ( rank `  A )  e. 
om ) )
13 peano2 4692 . . . . . . . 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 7547 . . . . . . . . . 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 3644 . . . . . . . . 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 4678 . . . . . . . . . 10  |-  ( (
rank `  A )  e.  om  ->  ( rank `  A )  e.  On )
20 r1suc 7458 . . . . . . . . . 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 2373 . . . . . . 7  |-  ( ( ( rank `  A
)  e.  om  /\  om  e.  On )  ->  A  e.  ( R1 ` 
suc  ( rank `  A
) ) )
24 fveq2 5541 . . . . . . . . 9  |-  ( x  =  suc  ( rank `  A )  ->  ( R1 `  x )  =  ( R1 `  suc  ( rank `  A )
) )
2524eleq2d 2363 . . . . . . . 8  |-  ( x  =  suc  ( rank `  A )  ->  ( A  e.  ( R1 `  x )  <->  A  e.  ( R1 `  suc  ( rank `  A ) ) ) )
2625rspcev 2897 . . . . . . 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 7480 . . . . . . 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 7483 . . . . . 6  |-  ( rank `  A )  e.  On
3331, 322th 230 . . . . 5  |-  ( E. x  e.  On  A  e.  ( R1 `  x
)  <->  ( rank `  A
)  e.  On )
34 rexeq 2750 . . . . . 6  |-  ( om  =  On  ->  ( E. x  e.  om  A  e.  ( R1 `  x )  <->  E. x  e.  On  A  e.  ( R1 `  x ) ) )
35 eleq2 2357 . . . . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   Oncon0 4408   suc csuc 4410   omcom 4672   ` cfv 5271   R1cr1 7450   rankcrnk 7451   Hf chf 24874
This theorem is referenced by:  elhf2g  24878  hfsn  24881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-r1 7452  df-rank 7453  df-hf 24875
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