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Theorem infpssALT 8149
Description: A set with a denumerable subset has a proper subset equinumerous to it, proved without AC or Infinity. Unlike infpss 8053, 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 8148 . 2  |-  -.  om  e. FinIV
2 reldom 7074 . . . . 5  |-  Rel  ~<_
32brrelex2i 4878 . . . 4  |-  ( om  ~<_  A  ->  A  e.  _V )
4 isfin4 8133 . . . 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 8147 . . . 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 1721   _Vcvv 2916    C. wpss 3281   class class class wbr 4172   omcom 4804    ~~ cen 7065    ~<_ cdom 7066  FinIVcfin4 8116
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069  df-dom 7070  df-fin4 8123
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