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Theorem cardalephex 7717
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 443 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  om  C_  A
)
2 cardaleph 7716 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) )
32sseq2d 3206 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) ) )
4 alephgeom 7709 . . . . . 6  |-  ( |^| { y  e.  On  |  A  C_  ( aleph `  y
) }  e.  On  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
53, 4syl6bbr 254 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
)
61, 5mpbid 201 . . . 4  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
7 fveq2 5525 . . . . . 6  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
87eqeq2d 2294 . . . . 5  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( A  =  ( aleph `  x )  <->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) ) )
98rspcev 2884 . . . 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 642 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  E. x  e.  On  A  =  (
aleph `  x ) )
1110ex 423 . 2  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  ->  E. x  e.  On  A  =  (
aleph `  x ) ) )
12 alephcard 7697 . . . 4  |-  ( card `  ( aleph `  x )
)  =  ( aleph `  x )
13 fveq2 5525 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  (
card `  ( aleph `  x
) ) )
14 id 19 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  A  =  ( aleph `  x )
)
1512, 13, 143eqtr4a 2341 . . 3  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1615rexlimivw 2663 . 2  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1711, 16impbid1 194 1  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  <->  E. x  e.  On  A  =  ( aleph `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547    C_ wss 3152   |^|cint 3862   Oncon0 4392   omcom 4656   ` cfv 5255   cardccrd 7568   alephcale 7569
This theorem is referenced by:  infenaleph  7718  isinfcard  7719  alephfp  7735  alephval3  7737  dfac12k  7773  alephval2  8194  winalim2  8318
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-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-rmo 2551  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-se 4353  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-isom 5264  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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