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Theorem onfin 7067
Description: An ordinal number is finite iff it is a natural number. Proposition 10.32 of [TakeutiZaring] p. 92. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
onfin  |-  ( A  e.  On  ->  ( A  e.  Fin  <->  A  e.  om ) )

Proof of Theorem onfin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isfi 6901 . 2  |-  ( A  e.  Fin  <->  E. x  e.  om  A  ~~  x
)
2 onomeneq 7066 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  om )  ->  ( A  ~~  x  <->  A  =  x ) )
3 eleq1a 2365 . . . . . 6  |-  ( x  e.  om  ->  ( A  =  x  ->  A  e.  om ) )
43adantl 452 . . . . 5  |-  ( ( A  e.  On  /\  x  e.  om )  ->  ( A  =  x  ->  A  e.  om ) )
52, 4sylbid 206 . . . 4  |-  ( ( A  e.  On  /\  x  e.  om )  ->  ( A  ~~  x  ->  A  e.  om )
)
65rexlimdva 2680 . . 3  |-  ( A  e.  On  ->  ( E. x  e.  om  A  ~~  x  ->  A  e.  om ) )
7 enrefg 6909 . . . 4  |-  ( A  e.  om  ->  A  ~~  A )
8 breq2 4043 . . . . 5  |-  ( x  =  A  ->  ( A  ~~  x  <->  A  ~~  A ) )
98rspcev 2897 . . . 4  |-  ( ( A  e.  om  /\  A  ~~  A )  ->  E. x  e.  om  A  ~~  x )
107, 9mpdan 649 . . 3  |-  ( A  e.  om  ->  E. x  e.  om  A  ~~  x
)
116, 10impbid1 194 . 2  |-  ( A  e.  On  ->  ( E. x  e.  om  A  ~~  x  <->  A  e.  om ) )
121, 11syl5bb 248 1  |-  ( A  e.  On  ->  ( A  e.  Fin  <->  A  e.  om ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   Oncon0 4408   omcom 4672    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  onfin2  7068  fin17  8036  isfin7-2  8038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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