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Theorem infpssALT 8128
Description: A set with a denumerable subset has a proper subset equinumerous to it, proved without AC or Infinity. Unlike infpss 8032, it uses Replacement. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 16-May-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
infpssALT  |-  ( om  ~<_  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
Distinct variable group:    x, A

Proof of Theorem infpssALT
StepHypRef Expression
1 ominf4 8127 . 2  |-  -.  om  e. FinIV
2 reldom 7053 . . . . 5  |-  Rel  ~<_
32brrelex2i 4861 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
4 isfin4 8112 . . . 4  |-  ( A  e.  _V  ->  ( A  e. FinIV 
<->  -.  E. x ( x  C.  A  /\  x  ~~  A ) ) )
53, 4syl 16 . . 3  |-  ( om  ~<_  A  ->  ( A  e. FinIV  <->  -. 
E. x ( x 
C.  A  /\  x  ~~  A ) ) )
6 domfin4 8126 . . . 4  |-  ( ( A  e. FinIV  /\  om  ~<_  A )  ->  om  e. FinIV )
76expcom 425 . . 3  |-  ( om  ~<_  A  ->  ( A  e. FinIV  ->  om  e. FinIV ) )
85, 7sylbird 227 . 2  |-  ( om  ~<_  A  ->  ( -.  E. x ( x  C.  A  /\  x  ~~  A
)  ->  om  e. FinIV )
)
91, 8mt3i 120 1  |-  ( om  ~<_  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    e. wcel 1717   _Vcvv 2901    C. wpss 3266   class class class wbr 4155   omcom 4787    ~~ cen 7044    ~<_ cdom 7045  FinIVcfin4 8095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-er 6843  df-en 7048  df-dom 7049  df-fin4 8102
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