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Theorem cardalephex 8510
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 458 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  om  C_  A
)
2 cardaleph 8509 . . . . . . 7  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) )
32sseq2d 3489 . . . . . 6  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) ) )
4 alephgeom 8502 . . . . . 6  |-  ( |^| { y  e.  On  |  A  C_  ( aleph `  y
) }  e.  On  <->  om  C_  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
53, 4syl6bbr 266 . . . . 5  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  ( om  C_  A  <->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
)
61, 5mpbid 213 . . . 4  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  e.  On )
7 fveq2 5872 . . . . . 6  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( aleph `  x
)  =  ( aleph ` 
|^| { y  e.  On  |  A  C_  ( aleph `  y ) } ) )
87eqeq2d 2434 . . . . 5  |-  ( x  =  |^| { y  e.  On  |  A  C_  ( aleph `  y ) }  ->  ( A  =  ( aleph `  x )  <->  A  =  ( aleph `  |^| { y  e.  On  |  A  C_  ( aleph `  y
) } ) ) )
98rspcev 3179 . . . 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 665 . . 3  |-  ( ( om  C_  A  /\  ( card `  A )  =  A )  ->  E. x  e.  On  A  =  (
aleph `  x ) )
1110ex 435 . 2  |-  ( om  C_  A  ->  ( (
card `  A )  =  A  ->  E. x  e.  On  A  =  (
aleph `  x ) ) )
12 alephcard 8490 . . . 4  |-  ( card `  ( aleph `  x )
)  =  ( aleph `  x )
13 fveq2 5872 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  (
card `  ( aleph `  x
) ) )
14 id 23 . . . 4  |-  ( A  =  ( aleph `  x
)  ->  A  =  ( aleph `  x )
)
1512, 13, 143eqtr4a 2487 . . 3  |-  ( A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1615rexlimivw 2912 . 2  |-  ( E. x  e.  On  A  =  ( aleph `  x
)  ->  ( card `  A )  =  A )
1711, 16impbid1 206 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 187    /\ wa 370    = wceq 1437    e. wcel 1867   E.wrex 2774   {crab 2777    C_ wss 3433   |^|cint 4249   Oncon0 5433   ` cfv 5592   omcom 6697   cardccrd 8359   alephcale 8360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-inf2 8137
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-se 4805  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-riota 6258  df-om 6698  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-oi 8016  df-har 8064  df-card 8363  df-aleph 8364
This theorem is referenced by:  infenaleph  8511  isinfcard  8512  alephfp  8528  alephval3  8530  dfac12k  8566  alephval2  8986  winalim2  9110
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