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Theorem sucinc2 6499
Description: Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)
Hypothesis
Ref Expression
sucinc.1  |-  F  =  ( z  e.  _V  |->  suc  z )
Assertion
Ref Expression
sucinc2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  C_  ( F `  B ) )
Distinct variable groups:    z, A    z, B
Allowed substitution hint:    F( z)

Proof of Theorem sucinc2
StepHypRef Expression
1 eloni 4406 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 ordsucss 4536 . . . . 5  |-  ( Ord 
B  ->  ( A  e.  B  ->  suc  A  C_  B ) )
31, 2syl 14 . . . 4  |-  ( B  e.  On  ->  ( A  e.  B  ->  suc 
A  C_  B )
)
43imp 124 . . 3  |-  ( ( B  e.  On  /\  A  e.  B )  ->  suc  A  C_  B
)
5 sssucid 4446 . . 3  |-  B  C_  suc  B
64, 5sstrdi 3191 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  suc  A  C_  suc  B )
7 onelon 4415 . . 3  |-  ( ( B  e.  On  /\  A  e.  B )  ->  A  e.  On )
8 elex 2771 . . . 4  |-  ( A  e.  On  ->  A  e.  _V )
9 sucexg 4530 . . . 4  |-  ( A  e.  On  ->  suc  A  e.  _V )
10 suceq 4433 . . . . 5  |-  ( z  =  A  ->  suc  z  =  suc  A )
11 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
1210, 11fvmptg 5633 . . . 4  |-  ( ( A  e.  _V  /\  suc  A  e.  _V )  ->  ( F `  A
)  =  suc  A
)
138, 9, 12syl2anc 411 . . 3  |-  ( A  e.  On  ->  ( F `  A )  =  suc  A )
147, 13syl 14 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  =  suc  A
)
15 elex 2771 . . . 4  |-  ( B  e.  On  ->  B  e.  _V )
16 sucexg 4530 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  _V )
17 suceq 4433 . . . . 5  |-  ( z  =  B  ->  suc  z  =  suc  B )
1817, 11fvmptg 5633 . . . 4  |-  ( ( B  e.  _V  /\  suc  B  e.  _V )  ->  ( F `  B
)  =  suc  B
)
1915, 16, 18syl2anc 411 . . 3  |-  ( B  e.  On  ->  ( F `  B )  =  suc  B )
2019adantr 276 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  B
)  =  suc  B
)
216, 14, 203sstr4d 3224 1  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  C_  ( F `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164   _Vcvv 2760    C_ wss 3153    |-> cmpt 4090   Ord word 4393   Oncon0 4394   suc csuc 4396   ` cfv 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262
This theorem is referenced by: (None)
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