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Theorem sucinc 6025
Description: Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.)
Hypothesis
Ref Expression
sucinc.1  |-  F  =  ( z  e.  _V  |->  suc  z )
Assertion
Ref Expression
sucinc  |-  A. x  x  C_  ( F `  x )
Distinct variable group:    x, z
Allowed substitution hints:    F( x, z)

Proof of Theorem sucinc
StepHypRef Expression
1 sssucid 4152 . . 3  |-  x  C_  suc  x
2 vex 2560 . . . 4  |-  x  e. 
_V
32sucex 4225 . . . 4  |-  suc  x  e.  _V
4 suceq 4139 . . . . 5  |-  ( z  =  x  ->  suc  z  =  suc  x )
5 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
64, 5fvmptg 5248 . . . 4  |-  ( ( x  e.  _V  /\  suc  x  e.  _V )  ->  ( F `  x
)  =  suc  x
)
72, 3, 6mp2an 402 . . 3  |-  ( F `
 x )  =  suc  x
81, 7sseqtr4i 2978 . 2  |-  x  C_  ( F `  x )
98ax-gen 1338 1  |-  A. x  x  C_  ( F `  x )
Colors of variables: wff set class
Syntax hints:   A.wal 1241    = wceq 1243    e. wcel 1393   _Vcvv 2557    C_ wss 2917    |-> cmpt 3818   suc csuc 4102   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910
This theorem is referenced by: (None)
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