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Theorem cardalephex 8272
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 8271 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) )
32sseq2d 3396 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) ) )
4 alephgeom 8264 . . . . . 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 5703 . . . . . 6  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
87eqeq2d 2454 . . . . 5  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( A  =  ( aleph `  x )  <->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) ) )
98rspcev 3085 . . . 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 8252 . . . 4  |-  ( card `  ( aleph `  x )
)  =  ( aleph `  x )
13 fveq2 5703 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  (
card `  ( aleph `  x
) ) )
14 id 22 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  A  =  ( aleph `  x )
)
1512, 13, 143eqtr4a 2501 . . 3  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1615rexlimivw 2849 . 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 1369    e. wcel 1756   E.wrex 2728   {crab 2731    C_ wss 3340   |^|cint 4140   Oncon0 4731   ` cfv 5430   omcom 6488   cardccrd 8117   alephcale 8118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-om 6489  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-oi 7736  df-har 7785  df-card 8121  df-aleph 8122
This theorem is referenced by:  infenaleph  8273  isinfcard  8274  alephfp  8290  alephval3  8292  dfac12k  8328  alephval2  8748  winalim2  8875
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