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Theorem onomeneq 6935
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 6934 . . . . . . . . 9  |-  ( B  e.  om  ->  -.  B  ~~  suc  B )
21ad2antlr 710 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  B  ~~  suc  B )
3 enen1 6886 . . . . . . . . 9  |-  ( A 
~~  B  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B ) )
43adantl 454 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B
) )
52, 4mtbird 294 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  A  ~~  suc  B )
6 peano2 4567 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  suc  B  e.  om )
7 sssucid 4362 . . . . . . . . . . . . . 14  |-  B  C_  suc  B
8 ssdomg 6793 . . . . . . . . . . . . . 14  |-  ( suc 
B  e.  om  ->  ( B  C_  suc  B  ->  B  ~<_  suc  B )
)
96, 7, 8ee10 1372 . . . . . . . . . . . . 13  |-  ( B  e.  om  ->  B  ~<_  suc  B )
10 endomtr 6804 . . . . . . . . . . . . 13  |-  ( ( A  ~~  B  /\  B  ~<_  suc  B )  ->  A  ~<_  suc  B )
119, 10sylan2 462 . . . . . . . . . . . 12  |-  ( ( A  ~~  B  /\  B  e.  om )  ->  A  ~<_  suc  B )
1211ancoms 441 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  ~~  B )  ->  A  ~<_  suc  B )
1312a1d 24 . . . . . . . . . 10  |-  ( ( B  e.  om  /\  A  ~~  B )  -> 
( om  C_  A  ->  A  ~<_  suc  B )
)
1413adantll 697 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  A  ~<_  suc  B )
)
15 ssel 3097 . . . . . . . . . . . . . . 15  |-  ( om  C_  A  ->  ( B  e.  om  ->  B  e.  A ) )
1615com12 29 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  ( om  C_  A  ->  B  e.  A ) )
1716adantr 453 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  B  e.  A ) )
18 eloni 4295 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  Ord  A )
19 ordelsuc 4502 . . . . . . . . . . . . . 14  |-  ( ( B  e.  om  /\  Ord  A )  ->  ( B  e.  A  <->  suc  B  C_  A ) )
2018, 19sylan2 462 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( B  e.  A  <->  suc 
B  C_  A )
)
2117, 20sylibd 207 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  suc  B  C_  A
) )
22 ssdomg 6793 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( suc  B  C_  A  ->  suc 
B  ~<_  A ) )
2322adantl 454 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( suc  B  C_  A  ->  suc  B  ~<_  A ) )
2421, 23syld 42 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2524ancoms 441 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2625adantr 453 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2714, 26jcad 521 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  ( A  ~<_  suc  B  /\  suc  B  ~<_  A ) ) )
28 sbth 6866 . . . . . . . 8  |-  ( ( A  ~<_  suc  B  /\  suc  B  ~<_  A )  ->  A  ~~  suc  B )
2927, 28syl6 31 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  A  ~~  suc  B
) )
305, 29mtod 170 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  om  C_  A
)
31 ordom 4556 . . . . . . . . 9  |-  Ord  om
32 ordtri1 4318 . . . . . . . . 9  |-  ( ( Ord  om  /\  Ord  A )  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3331, 18, 32sylancr 647 . . . . . . . 8  |-  ( A  e.  On  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3433con2bid 321 . . . . . . 7  |-  ( A  e.  On  ->  ( A  e.  om  <->  -.  om  C_  A
) )
3534ad2antrr 709 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om 
<->  -.  om  C_  A
) )
3630, 35mpbird 225 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  e.  om )
37 simplr 734 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  B  e.  om )
3836, 37jca 520 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om  /\  B  e.  om ) )
39 nneneq 6929 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
4039biimpa 472 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4138, 40sylancom 651 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4241ex 425 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
43 eqeng 6781 . . 3  |-  ( A  e.  On  ->  ( A  =  B  ->  A 
~~  B ) )
4443adantr 453 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
4542, 44impbid 185 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    C_ wss 3078   class class class wbr 3920   Ord word 4284   Oncon0 4285   suc csuc 4287   omcom 4547    ~~ cen 6746    ~<_ cdom 6747
This theorem is referenced by:  onfin  6936  ficardom  7478  finnisoeu  7624
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752
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