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Theorem suceloni 6633
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 4920 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  A  ->  x  C_  A ) )
2 elsn 4041 . . . . . . . . . 10  |-  ( x  e.  { A }  <->  x  =  A )
3 eqimss 3556 . . . . . . . . . 10  |-  ( x  =  A  ->  x  C_  A )
42, 3sylbi 195 . . . . . . . . 9  |-  ( x  e.  { A }  ->  x  C_  A )
54a1i 11 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  { A }  ->  x  C_  A
) )
61, 5orim12d 836 . . . . . . 7  |-  ( A  e.  On  ->  (
( x  e.  A  \/  x  e.  { A } )  ->  (
x  C_  A  \/  x  C_  A ) ) )
7 df-suc 4884 . . . . . . . . 9  |-  suc  A  =  ( A  u.  { A } )
87eleq2i 2545 . . . . . . . 8  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
9 elun 3645 . . . . . . . 8  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
108, 9bitr2i 250 . . . . . . 7  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  x  e.  suc  A )
11 oridm 514 . . . . . . 7  |-  ( ( x  C_  A  \/  x  C_  A )  <->  x  C_  A
)
126, 10, 113imtr3g 269 . . . . . 6  |-  ( A  e.  On  ->  (
x  e.  suc  A  ->  x  C_  A )
)
13 sssucid 4955 . . . . . 6  |-  A  C_  suc  A
14 sstr2 3511 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  suc  A  ->  x  C_  suc  A ) )
1512, 13, 14syl6mpi 62 . . . . 5  |-  ( A  e.  On  ->  (
x  e.  suc  A  ->  x  C_  suc  A ) )
1615ralrimiv 2876 . . . 4  |-  ( A  e.  On  ->  A. x  e.  suc  A x  C_  suc  A )
17 dftr3 4544 . . . 4  |-  ( Tr 
suc  A  <->  A. x  e.  suc  A x  C_  suc  A )
1816, 17sylibr 212 . . 3  |-  ( A  e.  On  ->  Tr  suc  A )
19 onss 6611 . . . . 5  |-  ( A  e.  On  ->  A  C_  On )
20 snssi 4171 . . . . 5  |-  ( A  e.  On  ->  { A }  C_  On )
2119, 20unssd 3680 . . . 4  |-  ( A  e.  On  ->  ( A  u.  { A } )  C_  On )
227, 21syl5eqss 3548 . . 3  |-  ( A  e.  On  ->  suc  A 
C_  On )
23 ordon 6603 . . . 4  |-  Ord  On
24 trssord 4895 . . . . 5  |-  ( ( Tr  suc  A  /\  suc  A  C_  On  /\  Ord  On )  ->  Ord  suc  A
)
25243exp 1195 . . . 4  |-  ( Tr 
suc  A  ->  ( suc 
A  C_  On  ->  ( Ord  On  ->  Ord  suc 
A ) ) )
2623, 25mpii 43 . . 3  |-  ( Tr 
suc  A  ->  ( suc 
A  C_  On  ->  Ord 
suc  A ) )
2718, 22, 26sylc 60 . 2  |-  ( A  e.  On  ->  Ord  suc 
A )
28 sucexg 6630 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
29 elong 4886 . . 3  |-  ( suc 
A  e.  _V  ->  ( suc  A  e.  On  <->  Ord 
suc  A ) )
3028, 29syl 16 . 2  |-  ( A  e.  On  ->  ( suc  A  e.  On  <->  Ord  suc  A
) )
3127, 30mpbird 232 1  |-  ( A  e.  On  ->  suc  A  e.  On )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    u. cun 3474    C_ wss 3476   {csn 4027   Tr wtr 4540   Ord word 4877   Oncon0 4878   suc csuc 4880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884
This theorem is referenced by:  ordsuc  6634  unon  6651  onsuci  6658  ordunisuc2  6664  ordzsl  6665  onzsl  6666  tfindsg  6680  dfom2  6687  findsg  6712  tfrlem12  7059  oasuc  7175  omsuc  7177  onasuc  7179  oacl  7186  oneo  7231  omeulem1  7232  omeulem2  7233  oeordi  7237  oeworde  7243  oelim2  7245  oelimcl  7250  oeeulem  7251  oeeui  7252  oaabs2  7295  omxpenlem  7619  card2inf  7982  cantnflt  8092  cantnflem1d  8108  cantnfltOLD  8122  cantnflem1dOLD  8131  cnfcom  8145  cnfcomOLD  8153  r1ordg  8197  bndrank  8260  r1pw  8264  r1pwOLD  8265  tcrank  8303  onssnum  8422  dfac12lem2  8525  cfsuc  8638  cfsmolem  8651  fin1a2lem1  8781  fin1a2lem2  8782  ttukeylem7  8896  alephreg  8958  gch2  9054  winainflem  9072  winalim2  9075  r1wunlim  9116  nqereu  9308  ontgval  29749  ontgsucval  29750  onsuctop  29751
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