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Theorem ordsucss 4287
Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
Assertion
Ref Expression
ordsucss  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )

Proof of Theorem ordsucss
StepHypRef Expression
1 ordtr 4172 . 2  |-  ( Ord 
B  ->  Tr  B
)
2 trss 3913 . . . . 5  |-  ( Tr  B  ->  ( A  e.  B  ->  A  C_  B ) )
3 snssi 3558 . . . . . 6  |-  ( A  e.  B  ->  { A }  C_  B )
43a1i 9 . . . . 5  |-  ( Tr  B  ->  ( A  e.  B  ->  { A }  C_  B ) )
52, 4jcad 301 . . . 4  |-  ( Tr  B  ->  ( A  e.  B  ->  ( A 
C_  B  /\  { A }  C_  B ) ) )
6 unss 3160 . . . 4  |-  ( ( A  C_  B  /\  { A }  C_  B
)  <->  ( A  u.  { A } )  C_  B )
75, 6syl6ib 159 . . 3  |-  ( Tr  B  ->  ( A  e.  B  ->  ( A  u.  { A }
)  C_  B )
)
8 df-suc 4165 . . . 4  |-  suc  A  =  ( A  u.  { A } )
98sseq1i 3036 . . 3  |-  ( suc 
A  C_  B  <->  ( A  u.  { A } ) 
C_  B )
107, 9syl6ibr 160 . 2  |-  ( Tr  B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
111, 10syl 14 1  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1436    u. cun 2984    C_ wss 2986   {csn 3425   Tr wtr 3904   Ord word 4156   suc csuc 4159
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-uni 3631  df-tr 3905  df-iord 4160  df-suc 4165
This theorem is referenced by:  ordelsuc  4288  tfrlemibfn  6028  tfr1onlembfn  6044  tfrcllembfn  6057  sucinc2  6142  nndomo  6513  prarloclemn  6979
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