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Theorem sucinc2 6207
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 4202 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 ordsucss 4321 . . . . 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 122 . . 3  |-  ( ( B  e.  On  /\  A  e.  B )  ->  suc  A  C_  B
)
5 sssucid 4242 . . 3  |-  B  C_  suc  B
64, 5syl6ss 3037 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  suc  A  C_  suc  B )
7 onelon 4211 . . 3  |-  ( ( B  e.  On  /\  A  e.  B )  ->  A  e.  On )
8 elex 2630 . . . 4  |-  ( A  e.  On  ->  A  e.  _V )
9 sucexg 4315 . . . 4  |-  ( A  e.  On  ->  suc  A  e.  _V )
10 suceq 4229 . . . . 5  |-  ( z  =  A  ->  suc  z  =  suc  A )
11 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
1210, 11fvmptg 5380 . . . 4  |-  ( ( A  e.  _V  /\  suc  A  e.  _V )  ->  ( F `  A
)  =  suc  A
)
138, 9, 12syl2anc 403 . . 3  |-  ( A  e.  On  ->  ( F `  A )  =  suc  A )
147, 13syl 14 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  =  suc  A
)
15 elex 2630 . . . 4  |-  ( B  e.  On  ->  B  e.  _V )
16 sucexg 4315 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  _V )
17 suceq 4229 . . . . 5  |-  ( z  =  B  ->  suc  z  =  suc  B )
1817, 11fvmptg 5380 . . . 4  |-  ( ( B  e.  _V  /\  suc  B  e.  _V )  ->  ( F `  B
)  =  suc  B
)
1915, 16, 18syl2anc 403 . . 3  |-  ( B  e.  On  ->  ( F `  B )  =  suc  B )
2019adantr 270 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  B
)  =  suc  B
)
216, 14, 203sstr4d 3069 1  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  C_  ( F `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2999    |-> cmpt 3899   Ord word 4189   Oncon0 4190   suc csuc 4192   ` cfv 5015
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023
This theorem is referenced by: (None)
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