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Theorem sucinc 6206
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 4242 . . 3  |-  x  C_  suc  x
2 vex 2622 . . . 4  |-  x  e. 
_V
32sucex 4316 . . . 4  |-  suc  x  e.  _V
4 suceq 4229 . . . . 5  |-  ( z  =  x  ->  suc  z  =  suc  x )
5 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
64, 5fvmptg 5380 . . . 4  |-  ( ( x  e.  _V  /\  suc  x  e.  _V )  ->  ( F `  x
)  =  suc  x
)
72, 3, 6mp2an 417 . . 3  |-  ( F `
 x )  =  suc  x
81, 7sseqtr4i 3059 . 2  |-  x  C_  ( F `  x )
98ax-gen 1383 1  |-  A. x  x  C_  ( F `  x )
Colors of variables: wff set class
Syntax hints:   A.wal 1287    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2999    |-> cmpt 3899   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-id 4120  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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