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Theorem ordsucss 4311
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 4196 . 2  |-  ( Ord 
B  ->  Tr  B
)
2 trss 3937 . . . . 5  |-  ( Tr  B  ->  ( A  e.  B  ->  A  C_  B ) )
3 snssi 3576 . . . . . 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 3172 . . . 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 4189 . . . 4  |-  suc  A  =  ( A  u.  { A } )
98sseq1i 3048 . . 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 1438    u. cun 2995    C_ wss 2997   {csn 3441   Tr wtr 3928   Ord word 4180   suc csuc 4183
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-uni 3649  df-tr 3929  df-iord 4184  df-suc 4189
This theorem is referenced by:  ordelsuc  4312  tfrlemibfn  6075  tfr1onlembfn  6091  tfrcllembfn  6104  sucinc2  6189  nndomo  6560  prarloclemn  7037
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