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Theorem suceloni 4756
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 4587 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  A  ->  x  C_  A ) )
2 elsn 3793 . . . . . . . . . 10  |-  ( x  e.  { A }  <->  x  =  A )
3 eqimss 3364 . . . . . . . . . 10  |-  ( x  =  A  ->  x  C_  A )
42, 3sylbi 188 . . . . . . . . 9  |-  ( x  e.  { A }  ->  x  C_  A )
54a1i 11 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  { A }  ->  x  C_  A
) )
61, 5orim12d 812 . . . . . . 7  |-  ( A  e.  On  ->  (
( x  e.  A  \/  x  e.  { A } )  ->  (
x  C_  A  \/  x  C_  A ) ) )
7 df-suc 4551 . . . . . . . . 9  |-  suc  A  =  ( A  u.  { A } )
87eleq2i 2472 . . . . . . . 8  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
9 elun 3452 . . . . . . . 8  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
108, 9bitr2i 242 . . . . . . 7  |-  ( ( x  e.  A  \/  x  e.  { A } )  <->  x  e.  suc  A )
11 oridm 501 . . . . . . 7  |-  ( ( x  C_  A  \/  x  C_  A )  <->  x  C_  A
)
126, 10, 113imtr3g 261 . . . . . 6  |-  ( A  e.  On  ->  (
x  e.  suc  A  ->  x  C_  A )
)
13 sssucid 4622 . . . . . 6  |-  A  C_  suc  A
14 sstr2 3319 . . . . . 6  |-  ( x 
C_  A  ->  ( A  C_  suc  A  ->  x  C_  suc  A ) )
1512, 13, 14syl6mpi 60 . . . . 5  |-  ( A  e.  On  ->  (
x  e.  suc  A  ->  x  C_  suc  A ) )
1615ralrimiv 2752 . . . 4  |-  ( A  e.  On  ->  A. x  e.  suc  A x  C_  suc  A )
17 dftr3 4270 . . . 4  |-  ( Tr 
suc  A  <->  A. x  e.  suc  A x  C_  suc  A )
1816, 17sylibr 204 . . 3  |-  ( A  e.  On  ->  Tr  suc  A )
19 onss 4734 . . . . 5  |-  ( A  e.  On  ->  A  C_  On )
20 snssi 3906 . . . . 5  |-  ( A  e.  On  ->  { A }  C_  On )
2119, 20unssd 3487 . . . 4  |-  ( A  e.  On  ->  ( A  u.  { A } )  C_  On )
227, 21syl5eqss 3356 . . 3  |-  ( A  e.  On  ->  suc  A 
C_  On )
23 ordon 4726 . . . 4  |-  Ord  On
24 trssord 4562 . . . . 5  |-  ( ( Tr  suc  A  /\  suc  A  C_  On  /\  Ord  On )  ->  Ord  suc  A
)
25243exp 1152 . . . 4  |-  ( Tr 
suc  A  ->  ( suc 
A  C_  On  ->  ( Ord  On  ->  Ord  suc 
A ) ) )
2623, 25mpii 41 . . 3  |-  ( Tr 
suc  A  ->  ( suc 
A  C_  On  ->  Ord 
suc  A ) )
2718, 22, 26sylc 58 . 2  |-  ( A  e.  On  ->  Ord  suc 
A )
28 sucexg 4753 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
29 elong 4553 . . 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 224 1  |-  ( A  e.  On  ->  suc  A  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920    u. cun 3282    C_ wss 3284   {csn 3778   Tr wtr 4266   Ord word 4544   Oncon0 4545   suc csuc 4547
This theorem is referenced by:  ordsuc  4757  unon  4774  onsuci  4781  ordunisuc2  4787  ordzsl  4788  onzsl  4789  tfindsg  4803  dfom2  4810  findsg  4835  tfrlem12  6613  oasuc  6731  omsuc  6733  onasuc  6735  oacl  6742  oneo  6787  omeulem1  6788  omeulem2  6789  oeordi  6793  oeworde  6799  oelim2  6801  oelimcl  6806  oeeulem  6807  oeeui  6808  oaabs2  6851  omxpenlem  7172  card2inf  7483  cantnflt  7587  cantnflem1d  7604  cnfcom  7617  r1ordg  7664  bndrank  7727  r1pw  7731  r1pwOLD  7732  tcrank  7768  onssnum  7881  dfac12lem2  7984  cfsuc  8097  cfsmolem  8110  fin1a2lem1  8240  fin1a2lem2  8241  ttukeylem7  8355  alephreg  8417  gch2  8514  winainflem  8528  winalim2  8531  r1wunlim  8572  nqereu  8766  ontgval  26089  ontgsucval  26090  onsuctop  26091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-tr 4267  df-eprel 4458  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-suc 4551
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