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Theorem suceloni 4793
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 4623 . . . . . . . 8  |-  ( A  e.  On  ->  (
x  e.  A  ->  x  C_  A ) )
2 elsn 3829 . . . . . . . . . 10  |-  ( x  e.  { A }  <->  x  =  A )
3 eqimss 3400 . . . . . . . . . 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 4587 . . . . . . . . 9  |-  suc  A  =  ( A  u.  { A } )
87eleq2i 2500 . . . . . . . 8  |-  ( x  e.  suc  A  <->  x  e.  ( A  u.  { A } ) )
9 elun 3488 . . . . . . . 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 4658 . . . . . 6  |-  A  C_  suc  A
14 sstr2 3355 . . . . . 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 2788 . . . 4  |-  ( A  e.  On  ->  A. x  e.  suc  A x  C_  suc  A )
17 dftr3 4306 . . . 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 4771 . . . . 5  |-  ( A  e.  On  ->  A  C_  On )
20 snssi 3942 . . . . 5  |-  ( A  e.  On  ->  { A }  C_  On )
2119, 20unssd 3523 . . . 4  |-  ( A  e.  On  ->  ( A  u.  { A } )  C_  On )
227, 21syl5eqss 3392 . . 3  |-  ( A  e.  On  ->  suc  A 
C_  On )
23 ordon 4763 . . . 4  |-  Ord  On
24 trssord 4598 . . . . 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 4790 . . 3  |-  ( A  e.  On  ->  suc  A  e.  _V )
29 elong 4589 . . 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 2705   _Vcvv 2956    u. cun 3318    C_ wss 3320   {csn 3814   Tr wtr 4302   Ord word 4580   Oncon0 4581   suc csuc 4583
This theorem is referenced by:  ordsuc  4794  unon  4811  onsuci  4818  ordunisuc2  4824  ordzsl  4825  onzsl  4826  tfindsg  4840  dfom2  4847  findsg  4872  tfrlem12  6650  oasuc  6768  omsuc  6770  onasuc  6772  oacl  6779  oneo  6824  omeulem1  6825  omeulem2  6826  oeordi  6830  oeworde  6836  oelim2  6838  oelimcl  6843  oeeulem  6844  oeeui  6845  oaabs2  6888  omxpenlem  7209  card2inf  7523  cantnflt  7627  cantnflem1d  7644  cnfcom  7657  r1ordg  7704  bndrank  7767  r1pw  7771  r1pwOLD  7772  tcrank  7808  onssnum  7921  dfac12lem2  8024  cfsuc  8137  cfsmolem  8150  fin1a2lem1  8280  fin1a2lem2  8281  ttukeylem7  8395  alephreg  8457  gch2  8554  winainflem  8568  winalim2  8571  r1wunlim  8612  nqereu  8806  ontgval  26181  ontgsucval  26182  onsuctop  26183
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-suc 4587
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