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Theorem onomeneq 7018
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 7017 . . . . . . . . 9  |-  ( B  e.  om  ->  -.  B  ~~  suc  B )
21ad2antlr 710 . . . . . . . 8  |-  ( ( ( A  e.  On  /\  B  e.  om )  /\  A  ~~  B )  ->  -.  B  ~~  suc  B )
3 enen1 6969 . . . . . . . . 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 4648 . . . . . . . . . . . . . 14  |-  ( B  e.  om  ->  suc  B  e.  om )
7 sssucid 4441 . . . . . . . . . . . . . 14  |-  B  C_  suc  B
8 ssdomg 6875 . . . . . . . . . . . . . 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 6887 . . . . . . . . . . . . 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 3149 . . . . . . . . . . . . . . 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 4374 . . . . . . . . . . . . . 14  |-  ( A  e.  On  ->  Ord  A )
19 ordelsuc 4583 . . . . . . . . . . . . . 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 6875 . . . . . . . . . . . . 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 6949 . . . . . . . 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 4637 . . . . . . . . 9  |-  Ord  om
32 ordtri1 4397 . . . . . . . . 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 7012 . . . . 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 6863 . . 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 3127   class class class wbr 3997   Ord word 4363   Oncon0 4364   suc csuc 4366   omcom 4628    ~~ cen 6828    ~<_ cdom 6829
This theorem is referenced by:  onfin  7019  ficardom  7562  finnisoeu  7708
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834
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