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Theorem onomeneq 7050
Description: An ordinal number equinumerous to a natural number is equal to it. Proposition 10.22 of [TakeutiZaring] p. 90 and its converse. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
onomeneq  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )

Proof of Theorem onomeneq
StepHypRef Expression
1 php5 7049 . . . . . . . . 9  |-  ( B  e.  om  ->  -.  B  ~~  suc  B )
21ad2antlr 707 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  B  ~~  suc  B )
3 enen1 7001 . . . . . . . . 9  |-  ( A 
~~  B  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B ) )
43adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B
) )
52, 4mtbird 292 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  A  ~~  suc  B )
6 peano2 4676 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  suc  B  e.  om )
7 sssucid 4469 . . . . . . . . . . . . . 14  |-  B  C_  suc  B
8 ssdomg 6907 . . . . . . . . . . . . . 14  |-  ( suc 
B  e.  om  ->  ( B  C_  suc  B  ->  B  ~<_  suc  B )
)
96, 7, 8ee10 1366 . . . . . . . . . . . . 13  |-  ( B  e.  om  ->  B  ~<_  suc  B )
10 endomtr 6919 . . . . . . . . . . . . 13  |-  ( ( A  ~~  B  /\  B  ~<_  suc  B )  ->  A  ~<_  suc  B )
119, 10sylan2 460 . . . . . . . . . . . 12  |-  ( ( A  ~~  B  /\  B  e.  om )  ->  A  ~<_  suc  B )
1211ancoms 439 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  ~~  B )  ->  A  ~<_  suc  B )
1312a1d 22 . . . . . . . . . 10  |-  ( ( B  e.  om  /\  A  ~~  B )  -> 
( om  C_  A  ->  A  ~<_  suc  B )
)
1413adantll 694 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  A  ~<_  suc  B )
)
15 ssel 3174 . . . . . . . . . . . . . . 15  |-  ( om  C_  A  ->  ( B  e.  om  ->  B  e.  A ) )
1615com12 27 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  ( om  C_  A  ->  B  e.  A ) )
1716adantr 451 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  B  e.  A ) )
18 eloni 4402 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  Ord  A )
19 ordelsuc 4611 . . . . . . . . . . . . . 14  |-  ( ( B  e.  om  /\  Ord  A )  ->  ( B  e.  A  <->  suc  B  C_  A ) )
2018, 19sylan2 460 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( B  e.  A  <->  suc 
B  C_  A )
)
2117, 20sylibd 205 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  suc  B  C_  A
) )
22 ssdomg 6907 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( suc  B  C_  A  ->  suc 
B  ~<_  A ) )
2322adantl 452 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( suc  B  C_  A  ->  suc  B  ~<_  A ) )
2421, 23syld 40 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2524ancoms 439 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2625adantr 451 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2714, 26jcad 519 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  ( A  ~<_  suc  B  /\  suc  B  ~<_  A ) ) )
28 sbth 6981 . . . . . . . 8  |-  ( ( A  ~<_  suc  B  /\  suc  B  ~<_  A )  ->  A  ~~  suc  B )
2927, 28syl6 29 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  A  ~~  suc  B
) )
305, 29mtod 168 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  om  C_  A
)
31 ordom 4665 . . . . . . . . 9  |-  Ord  om
32 ordtri1 4425 . . . . . . . . 9  |-  ( ( Ord  om  /\  Ord  A )  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3331, 18, 32sylancr 644 . . . . . . . 8  |-  ( A  e.  On  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3433con2bid 319 . . . . . . 7  |-  ( A  e.  On  ->  ( A  e.  om  <->  -.  om  C_  A
) )
3534ad2antrr 706 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om 
<->  -.  om  C_  A
) )
3630, 35mpbird 223 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  e.  om )
37 simplr 731 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  B  e.  om )
3836, 37jca 518 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om  /\  B  e.  om ) )
39 nneneq 7044 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
4039biimpa 470 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4138, 40sylancom 648 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4241ex 423 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
43 eqeng 6895 . . 3  |-  ( A  e.  On  ->  ( A  =  B  ->  A 
~~  B ) )
4443adantr 451 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
4542, 44impbid 183 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   Ord word 4391   Oncon0 4392   suc csuc 4394   omcom 4656    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  onfin  7051  ficardom  7594  finnisoeu  7740
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  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-er 6660  df-en 6864  df-dom 6865  df-sdom 6866
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