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Theorem cardalephex 7960
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 444 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  om  C_  A
)
2 cardaleph 7959 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) )
32sseq2d 3368 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) ) )
4 alephgeom 7952 . . . . . 6  |-  ( |^| { y  e.  On  |  A  C_  ( aleph `  y
) }  e.  On  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
53, 4syl6bbr 255 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
)
61, 5mpbid 202 . . . 4  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
7 fveq2 5719 . . . . . 6  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
87eqeq2d 2446 . . . . 5  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( A  =  ( aleph `  x )  <->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) ) )
98rspcev 3044 . . . 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 643 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  E. x  e.  On  A  =  (
aleph `  x ) )
1110ex 424 . 2  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  ->  E. x  e.  On  A  =  (
aleph `  x ) ) )
12 alephcard 7940 . . . 4  |-  ( card `  ( aleph `  x )
)  =  ( aleph `  x )
13 fveq2 5719 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  (
card `  ( aleph `  x
) ) )
14 id 20 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  A  =  ( aleph `  x )
)
1512, 13, 143eqtr4a 2493 . . 3  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1615rexlimivw 2818 . 2  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1711, 16impbid1 195 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701    C_ wss 3312   |^|cint 4042   Oncon0 4573   omcom 4836   ` cfv 5445   cardccrd 7811   alephcale 7812
This theorem is referenced by:  infenaleph  7961  isinfcard  7962  alephfp  7978  alephval3  7980  dfac12k  8016  alephval2  8436  winalim2  8560
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-riota 6540  df-recs 6624  df-rdg 6659  df-er 6896  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816
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