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Theorem sucinc 6056
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 4180 . . 3  |-  x  C_  suc  x
2 vex 2577 . . . 4  |-  x  e. 
_V
32sucex 4253 . . . 4  |-  suc  x  e.  _V
4 suceq 4167 . . . . 5  |-  ( z  =  x  ->  suc  z  =  suc  x )
5 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
64, 5fvmptg 5276 . . . 4  |-  ( ( x  e.  _V  /\  suc  x  e.  _V )  ->  ( F `  x
)  =  suc  x
)
72, 3, 6mp2an 410 . . 3  |-  ( F `
 x )  =  suc  x
81, 7sseqtr4i 3006 . 2  |-  x  C_  ( F `  x )
98ax-gen 1354 1  |-  A. x  x  C_  ( F `  x )
Colors of variables: wff set class
Syntax hints:   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574    C_ wss 2945    |-> cmpt 3846   suc csuc 4130   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938
This theorem is referenced by: (None)
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