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Theorem sucinc 6246
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 4266 . . 3  |-  x  C_  suc  x
2 vex 2636 . . . 4  |-  x  e. 
_V
32sucex 4344 . . . 4  |-  suc  x  e.  _V
4 suceq 4253 . . . . 5  |-  ( z  =  x  ->  suc  z  =  suc  x )
5 sucinc.1 . . . . 5  |-  F  =  ( z  e.  _V  |->  suc  z )
64, 5fvmptg 5415 . . . 4  |-  ( ( x  e.  _V  /\  suc  x  e.  _V )  ->  ( F `  x
)  =  suc  x
)
72, 3, 6mp2an 418 . . 3  |-  ( F `
 x )  =  suc  x
81, 7sseqtr4i 3074 . 2  |-  x  C_  ( F `  x )
98ax-gen 1390 1  |-  A. x  x  C_  ( F `  x )
Colors of variables: wff set class
Syntax hints:   A.wal 1294    = wceq 1296    e. wcel 1445   _Vcvv 2633    C_ wss 3013    |-> cmpt 3921   suc csuc 4216   ` cfv 5049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057
This theorem is referenced by: (None)
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