MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suceloni Structured version   Unicode version

Theorem suceloni 4785
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 4615 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  A  ->  x  C_  A ) )
2 elsn 3821 . . . . . . . . . 10  |-  ( x  e.  { A }  <->  x  =  A )
3 eqimss 3392 . . . . . . . . . 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 4579 . . . . . . . . 9  |-  suc  A  =  ( A  u.  { A } )
87eleq2i 2499 . . . . . . . 8  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
9 elun 3480 . . . . . . . 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 4650 . . . . . 6  |-  A  C_  suc  A
14 sstr2 3347 . . . . . 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 2780 . . . 4  |-  ( A  e.  On  ->  A. x  e.  suc  A x  C_  suc  A )
17 dftr3 4298 . . . 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 4763 . . . . 5  |-  ( A  e.  On  ->  A  C_  On )
20 snssi 3934 . . . . 5  |-  ( A  e.  On  ->  { A }  C_  On )
2119, 20unssd 3515 . . . 4  |-  ( A  e.  On  ->  ( A  u.  { A } )  C_  On )
227, 21syl5eqss 3384 . . 3  |-  ( A  e.  On  ->  suc  A 
C_  On )
23 ordon 4755 . . . 4  |-  Ord  On
24 trssord 4590 . . . . 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 4782 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
29 elong 4581 . . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    u. cun 3310    C_ wss 3312   {csn 3806   Tr wtr 4294   Ord word 4572   Oncon0 4573   suc csuc 4575
This theorem is referenced by:  ordsuc  4786  unon  4803  onsuci  4810  ordunisuc2  4816  ordzsl  4817  onzsl  4818  tfindsg  4832  dfom2  4839  findsg  4864  tfrlem12  6642  oasuc  6760  omsuc  6762  onasuc  6764  oacl  6771  oneo  6816  omeulem1  6817  omeulem2  6818  oeordi  6822  oeworde  6828  oelim2  6830  oelimcl  6835  oeeulem  6836  oeeui  6837  oaabs2  6880  omxpenlem  7201  card2inf  7515  cantnflt  7619  cantnflem1d  7636  cnfcom  7649  r1ordg  7696  bndrank  7759  r1pw  7763  r1pwOLD  7764  tcrank  7800  onssnum  7913  dfac12lem2  8016  cfsuc  8129  cfsmolem  8142  fin1a2lem1  8272  fin1a2lem2  8273  ttukeylem7  8387  alephreg  8449  gch2  8546  winainflem  8560  winalim2  8563  r1wunlim  8604  nqereu  8798  ontgval  26173  ontgsucval  26174  onsuctop  26175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-suc 4579
  Copyright terms: Public domain W3C validator