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Theorem cardalephex 8472
Description: Every transfinite cardinal is an aleph and vice-versa. Theorem 8A(b) of [Enderton] p. 213 and its converse. (Contributed by NM, 5-Nov-2003.)
Assertion
Ref Expression
cardalephex  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
Distinct variable group:    x, A

Proof of Theorem cardalephex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  om  C_  A
)
2 cardaleph 8471 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) )
32sseq2d 3532 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) ) )
4 alephgeom 8464 . . . . . 6  |-  ( |^| { y  e.  On  |  A  C_  ( aleph `  y
) }  e.  On  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
53, 4syl6bbr 263 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
)
61, 5mpbid 210 . . . 4  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
7 fveq2 5866 . . . . . 6  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
87eqeq2d 2481 . . . . 5  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( A  =  ( aleph `  x )  <->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) ) )
98rspcev 3214 . . . 4  |-  ( (
|^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On  /\  A  =  (
aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )  ->  E. x  e.  On  A  =  ( aleph `  x ) )
106, 2, 9syl2anc 661 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  E. x  e.  On  A  =  (
aleph `  x ) )
1110ex 434 . 2  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  ->  E. x  e.  On  A  =  (
aleph `  x ) ) )
12 alephcard 8452 . . . 4  |-  ( card `  ( aleph `  x )
)  =  ( aleph `  x )
13 fveq2 5866 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  (
card `  ( aleph `  x
) ) )
14 id 22 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  A  =  ( aleph `  x )
)
1512, 13, 143eqtr4a 2534 . . 3  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1615rexlimivw 2952 . 2  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1711, 16impbid1 203 1  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818    C_ wss 3476   |^|cint 4282   Oncon0 4878   ` cfv 5588   omcom 6685   cardccrd 8317   alephcale 8318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-inf2 8059
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6246  df-om 6686  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-oi 7936  df-har 7985  df-card 8321  df-aleph 8322
This theorem is referenced by:  infenaleph  8473  isinfcard  8474  alephfp  8490  alephval3  8492  dfac12k  8528  alephval2  8948  winalim2  9075
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