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Theorem onomeneq 7665
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 7663 . . . . . . . . 9  |-  ( B  e.  om  ->  -.  B  ~~  suc  B )
21ad2antlr 725 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  B  ~~  suc  B )
3 enen1 7615 . . . . . . . . 9  |-  ( A 
~~  B  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B ) )
43adantl 464 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B
) )
52, 4mtbird 299 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  A  ~~  suc  B )
6 peano2 6658 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  suc  B  e.  om )
7 sssucid 4898 . . . . . . . . . . . . . 14  |-  B  C_  suc  B
8 ssdomg 7519 . . . . . . . . . . . . . 14  |-  ( suc 
B  e.  om  ->  ( B  C_  suc  B  ->  B  ~<_  suc  B )
)
96, 7, 8mpisyl 19 . . . . . . . . . . . . 13  |-  ( B  e.  om  ->  B  ~<_  suc  B )
10 endomtr 7531 . . . . . . . . . . . . 13  |-  ( ( A  ~~  B  /\  B  ~<_  suc  B )  ->  A  ~<_  suc  B )
119, 10sylan2 472 . . . . . . . . . . . 12  |-  ( ( A  ~~  B  /\  B  e.  om )  ->  A  ~<_  suc  B )
1211ancoms 451 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  ~~  B )  ->  A  ~<_  suc  B )
1312a1d 25 . . . . . . . . . 10  |-  ( ( B  e.  om  /\  A  ~~  B )  -> 
( om  C_  A  ->  A  ~<_  suc  B )
)
1413adantll 712 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  A  ~<_  suc  B )
)
15 ssel 3435 . . . . . . . . . . . . . . 15  |-  ( om  C_  A  ->  ( B  e.  om  ->  B  e.  A ) )
1615com12 29 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  ( om  C_  A  ->  B  e.  A ) )
1716adantr 463 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  B  e.  A ) )
18 eloni 4831 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  Ord  A )
19 ordelsuc 6593 . . . . . . . . . . . . . 14  |-  ( ( B  e.  om  /\  Ord  A )  ->  ( B  e.  A  <->  suc  B  C_  A ) )
2018, 19sylan2 472 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( B  e.  A  <->  suc 
B  C_  A )
)
2117, 20sylibd 214 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  suc  B  C_  A
) )
22 ssdomg 7519 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( suc  B  C_  A  ->  suc 
B  ~<_  A ) )
2322adantl 464 . . . . . . . . . . . 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 451 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2625adantr 463 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2714, 26jcad 531 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  ( A  ~<_  suc  B  /\  suc  B  ~<_  A ) ) )
28 sbth 7595 . . . . . . . 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 177 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  om  C_  A
)
31 ordom 6647 . . . . . . . . 9  |-  Ord  om
32 ordtri1 4854 . . . . . . . . 9  |-  ( ( Ord  om  /\  Ord  A )  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3331, 18, 32sylancr 661 . . . . . . . 8  |-  ( A  e.  On  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3433con2bid 327 . . . . . . 7  |-  ( A  e.  On  ->  ( A  e.  om  <->  -.  om  C_  A
) )
3534ad2antrr 724 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om 
<->  -.  om  C_  A
) )
3630, 35mpbird 232 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  e.  om )
37 simplr 754 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  B  e.  om )
3836, 37jca 530 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om  /\  B  e.  om ) )
39 nneneq 7658 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
4039biimpa 482 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4138, 40sylancom 665 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4241ex 432 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
43 eqeng 7507 . . 3  |-  ( A  e.  On  ->  ( A  =  B  ->  A 
~~  B ) )
4443adantr 463 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  =  B  ->  A  ~~  B
) )
4542, 44impbid 191 1  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3413   class class class wbr 4394   Ord word 4820   Oncon0 4821   suc csuc 4823   omcom 6638    ~~ cen 7471    ~<_ cdom 7472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-om 6639  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477
This theorem is referenced by:  onfin  7666  ficardom  8294  finnisoeu  8446
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