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Theorem ordsucss 4380
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 4260 . 2  |-  ( Ord 
B  ->  Tr  B
)
2 trss 3995 . . . . 5  |-  ( Tr  B  ->  ( A  e.  B  ->  A  C_  B ) )
3 snssi 3630 . . . . . 6  |-  ( A  e.  B  ->  { A }  C_  B )
43a1i 9 . . . . 5  |-  ( Tr  B  ->  ( A  e.  B  ->  { A }  C_  B ) )
52, 4jcad 303 . . . 4  |-  ( Tr  B  ->  ( A  e.  B  ->  ( A 
C_  B  /\  { A }  C_  B ) ) )
6 unss 3216 . . . 4  |-  ( ( A  C_  B  /\  { A }  C_  B
)  <->  ( A  u.  { A } )  C_  B )
75, 6syl6ib 160 . . 3  |-  ( Tr  B  ->  ( A  e.  B  ->  ( A  u.  { A }
)  C_  B )
)
8 df-suc 4253 . . . 4  |-  suc  A  =  ( A  u.  { A } )
98sseq1i 3089 . . 3  |-  ( suc 
A  C_  B  <->  ( A  u.  { A } ) 
C_  B )
107, 9syl6ibr 161 . 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 103    e. wcel 1463    u. cun 3035    C_ wss 3037   {csn 3493   Tr wtr 3986   Ord word 4244   suc csuc 4247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-uni 3703  df-tr 3987  df-iord 4248  df-suc 4253
This theorem is referenced by:  ordelsuc  4381  tfrlemibfn  6179  tfr1onlembfn  6195  tfrcllembfn  6208  sucinc2  6296  nndomo  6711  prarloclemn  7255  ennnfonelemhom  11773  ennnfonelemrn  11777
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