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Theorem onomeneq 7614
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 7612 . . . . . . . . 9  |-  ( B  e.  om  ->  -.  B  ~~  suc  B )
21ad2antlr 726 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  B  ~~  suc  B )
3 enen1 7564 . . . . . . . . 9  |-  ( A 
~~  B  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B ) )
43adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  ~~  suc  B  <->  B  ~~  suc  B
) )
52, 4mtbird 301 . . . . . . 7  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  A  ~~  suc  B )
6 peano2 6609 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  suc  B  e.  om )
7 sssucid 4907 . . . . . . . . . . . . . 14  |-  B  C_  suc  B
8 ssdomg 7468 . . . . . . . . . . . . . 14  |-  ( suc 
B  e.  om  ->  ( B  C_  suc  B  ->  B  ~<_  suc  B )
)
96, 7, 8mpisyl 18 . . . . . . . . . . . . 13  |-  ( B  e.  om  ->  B  ~<_  suc  B )
10 endomtr 7480 . . . . . . . . . . . . 13  |-  ( ( A  ~~  B  /\  B  ~<_  suc  B )  ->  A  ~<_  suc  B )
119, 10sylan2 474 . . . . . . . . . . . 12  |-  ( ( A  ~~  B  /\  B  e.  om )  ->  A  ~<_  suc  B )
1211ancoms 453 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  ~~  B )  ->  A  ~<_  suc  B )
1312a1d 25 . . . . . . . . . 10  |-  ( ( B  e.  om  /\  A  ~~  B )  -> 
( om  C_  A  ->  A  ~<_  suc  B )
)
1413adantll 713 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  A  ~<_  suc  B )
)
15 ssel 3461 . . . . . . . . . . . . . . 15  |-  ( om  C_  A  ->  ( B  e.  om  ->  B  e.  A ) )
1615com12 31 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  ( om  C_  A  ->  B  e.  A ) )
1716adantr 465 . . . . . . . . . . . . 13  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  B  e.  A ) )
18 eloni 4840 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  Ord  A )
19 ordelsuc 6544 . . . . . . . . . . . . . 14  |-  ( ( B  e.  om  /\  Ord  A )  ->  ( B  e.  A  <->  suc  B  C_  A ) )
2018, 19sylan2 474 . . . . . . . . . . . . 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 7468 . . . . . . . . . . . . 13  |-  ( A  e.  On  ->  ( suc  B  C_  A  ->  suc 
B  ~<_  A ) )
2322adantl 466 . . . . . . . . . . . 12  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( suc  B  C_  A  ->  suc  B  ~<_  A ) )
2421, 23syld 44 . . . . . . . . . . 11  |-  ( ( B  e.  om  /\  A  e.  On )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2524ancoms 453 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2625adantr 465 . . . . . . . . 9  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  suc  B  ~<_  A ) )
2714, 26jcad 533 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( om  C_  A  ->  ( A  ~<_  suc  B  /\  suc  B  ~<_  A ) ) )
28 sbth 7544 . . . . . . . 8  |-  ( ( A  ~<_  suc  B  /\  suc  B  ~<_  A )  ->  A  ~~  suc  B )
2927, 28syl6 33 . . . . . . 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 6598 . . . . . . . . 9  |-  Ord  om
32 ordtri1 4863 . . . . . . . . 9  |-  ( ( Ord  om  /\  Ord  A )  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3331, 18, 32sylancr 663 . . . . . . . 8  |-  ( A  e.  On  ->  ( om  C_  A  <->  -.  A  e.  om ) )
3433con2bid 329 . . . . . . 7  |-  ( A  e.  On  ->  ( A  e.  om  <->  -.  om  C_  A
) )
3534ad2antrr 725 . . . . . 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 532 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  ( A  e. 
om  /\  B  e.  om ) )
39 nneneq 7607 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  ~~  B  <->  A  =  B ) )
4039biimpa 484 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4138, 40sylancom 667 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  A  =  B )
4241ex 434 . 2  |-  ( ( A  e.  On  /\  B  e.  om )  ->  ( A  ~~  B  ->  A  =  B ) )
43 eqeng 7456 . . 3  |-  ( A  e.  On  ->  ( A  =  B  ->  A 
~~  B ) )
4443adantr 465 . 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 369    = wceq 1370    e. wcel 1758    C_ wss 3439   class class class wbr 4403   Ord word 4829   Oncon0 4830   suc csuc 4832   omcom 6589    ~~ cen 7420    ~<_ cdom 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-om 6590  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426
This theorem is referenced by:  onfin  7615  ficardom  8245  finnisoeu  8397
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