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Theorem suceloni 4604
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
suceloni  |-  ( A  e.  On  ->  suc  A  e.  On )

Proof of Theorem suceloni
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 onelss 4434 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  A  ->  x  C_  A ) )
2 elsn 3655 . . . . . . . . . 10  |-  ( x  e.  { A }  <->  x  =  A )
3 eqimss 3230 . . . . . . . . . 10  |-  ( x  =  A  ->  x  C_  A )
42, 3sylbi 187 . . . . . . . . 9  |-  ( x  e.  { A }  ->  x  C_  A )
54a1i 10 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  { A }  ->  x  C_  A
) )
61, 5orim12d 811 . . . . . . 7  |-  ( A  e.  On  ->  (
( x  e.  A  \/  x  e.  { A } )  ->  (
x  C_  A  \/  x  C_  A ) ) )
7 df-suc 4398 . . . . . . . . 9  |-  suc  A  =  ( A  u.  { A } )
87eleq2i 2347 . . . . . . . 8  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
9 elun 3316 . . . . . . . 8  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
108, 9bitr2i 241 . . . . . . 7  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  x  e.  suc  A )
11 oridm 500 . . . . . . 7  |-  ( ( x  C_  A  \/  x  C_  A )  <->  x  C_  A
)
126, 10, 113imtr3g 260 . . . . . 6  |-  ( A  e.  On  ->  (
x  e.  suc  A  ->  x  C_  A )
)
13 sssucid 4469 . . . . . 6  |-  A  C_  suc  A
14 sstr2 3186 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  suc  A  ->  x  C_  suc  A ) )
1512, 13, 14syl6mpi 58 . . . . 5  |-  ( A  e.  On  ->  (
x  e.  suc  A  ->  x  C_  suc  A ) )
1615ralrimiv 2625 . . . 4  |-  ( A  e.  On  ->  A. x  e.  suc  A x  C_  suc  A )
17 dftr3 4117 . . . 4  |-  ( Tr 
suc  A  <->  A. x  e.  suc  A x  C_  suc  A )
1816, 17sylibr 203 . . 3  |-  ( A  e.  On  ->  Tr  suc  A )
19 onss 4582 . . . . 5  |-  ( A  e.  On  ->  A  C_  On )
20 snssi 3759 . . . . 5  |-  ( A  e.  On  ->  { A }  C_  On )
2119, 20unssd 3351 . . . 4  |-  ( A  e.  On  ->  ( A  u.  { A } )  C_  On )
227, 21syl5eqss 3222 . . 3  |-  ( A  e.  On  ->  suc  A 
C_  On )
23 ordon 4574 . . . 4  |-  Ord  On
24 trssord 4409 . . . . 5  |-  ( ( Tr  suc  A  /\  suc  A  C_  On  /\  Ord  On )  ->  Ord  suc  A
)
25243exp 1150 . . . 4  |-  ( Tr 
suc  A  ->  ( suc 
A  C_  On  ->  ( Ord  On  ->  Ord  suc 
A ) ) )
2623, 25mpii 39 . . 3  |-  ( Tr 
suc  A  ->  ( suc 
A  C_  On  ->  Ord 
suc  A ) )
2718, 22, 26sylc 56 . 2  |-  ( A  e.  On  ->  Ord  suc 
A )
28 sucexg 4601 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
29 elong 4400 . . 3  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  On  <->  Ord 
suc  A ) )
3028, 29syl 15 . 2  |-  ( A  e.  On  ->  ( suc  A  e.  On  <->  Ord  suc  A
) )
3127, 30mpbird 223 1  |-  ( A  e.  On  ->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    u. cun 3150    C_ wss 3152   {csn 3640   Tr wtr 4113   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  ordsuc  4605  unon  4622  onsuci  4629  ordunisuc2  4635  ordzsl  4636  onzsl  4637  tfindsg  4651  dfom2  4658  findsg  4683  tfrlem12  6405  oasuc  6523  omsuc  6525  onasuc  6527  oacl  6534  oneo  6579  omeulem1  6580  omeulem2  6581  oeordi  6585  oeworde  6591  oelim2  6593  oelimcl  6598  oeeulem  6599  oeeui  6600  oaabs2  6643  omxpenlem  6963  card2inf  7269  cantnflt  7373  cantnflem1d  7390  cnfcom  7403  r1ordg  7450  bndrank  7513  r1pw  7517  r1pwOLD  7518  tcrank  7554  onssnum  7667  dfac12lem2  7770  cfsuc  7883  cfsmolem  7896  fin1a2lem1  8026  fin1a2lem2  8027  ttukeylem7  8142  alephreg  8204  gch2  8301  winainflem  8315  winalim2  8318  r1wunlim  8359  nqereu  8553  ontgval  24870  ontgsucval  24871  onsuctop  24872
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-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-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-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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