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Theorem sucinc2 6422
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 4358 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 ordsucss 4486 . . . . 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 123 . . 3  |-  ( ( B  e.  On  /\  A  e.  B )  ->  suc  A  C_  B
)
5 sssucid 4398 . . 3  |-  B  C_  suc  B
64, 5sstrdi 3159 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  suc  A  C_  suc  B )
7 onelon 4367 . . 3  |-  ( ( B  e.  On  /\  A  e.  B )  ->  A  e.  On )
8 elex 2741 . . . 4  |-  ( A  e.  On  ->  A  e.  _V )
9 sucexg 4480 . . . 4  |-  ( A  e.  On  ->  suc  A  e.  _V )
10 suceq 4385 . . . . 5  |-  ( z  =  A  ->  suc  z  =  suc  A )
11 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
1210, 11fvmptg 5570 . . . 4  |-  ( ( A  e.  _V  /\  suc  A  e.  _V )  ->  ( F `  A
)  =  suc  A
)
138, 9, 12syl2anc 409 . . 3  |-  ( A  e.  On  ->  ( F `  A )  =  suc  A )
147, 13syl 14 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  =  suc  A
)
15 elex 2741 . . . 4  |-  ( B  e.  On  ->  B  e.  _V )
16 sucexg 4480 . . . 4  |-  ( B  e.  On  ->  suc  B  e.  _V )
17 suceq 4385 . . . . 5  |-  ( z  =  B  ->  suc  z  =  suc  B )
1817, 11fvmptg 5570 . . . 4  |-  ( ( B  e.  _V  /\  suc  B  e.  _V )  ->  ( F `  B
)  =  suc  B
)
1915, 16, 18syl2anc 409 . . 3  |-  ( B  e.  On  ->  ( F `  B )  =  suc  B )
2019adantr 274 . 2  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  B
)  =  suc  B
)
216, 14, 203sstr4d 3192 1  |-  ( ( B  e.  On  /\  A  e.  B )  ->  ( F `  A
)  C_  ( F `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141   _Vcvv 2730    C_ wss 3121    |-> cmpt 4048   Ord word 4345   Oncon0 4346   suc csuc 4348   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204
This theorem is referenced by: (None)
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