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Theorem infpss 7843
Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 7939. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infpss  |-  ( om  ~<_  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
Distinct variable group:    x, A

Proof of Theorem infpss
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 infn0 7119 . . 3  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2 n0 3464 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
31, 2sylib 188 . 2  |-  ( om  ~<_  A  ->  E. y 
y  e.  A )
4 reldom 6869 . . . . . . . 8  |-  Rel  ~<_
54brrelex2i 4730 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
6 difexg 4162 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  \  { y } )  e.  _V )
75, 6syl 15 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  \  { y } )  e.  _V )
87adantr 451 . . . . 5  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  e.  _V )
9 simpr 447 . . . . . . 7  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  y  e.  A )
10 disjsn 3693 . . . . . . . . 9  |-  ( ( A  i^i  { y } )  =  (/)  <->  -.  y  e.  A )
11 disj4 3503 . . . . . . . . 9  |-  ( ( A  i^i  { y } )  =  (/)  <->  -.  ( A  \  { y } )  C.  A
)
1210, 11bitr3i 242 . . . . . . . 8  |-  ( -.  y  e.  A  <->  -.  ( A  \  { y } )  C.  A )
1312con4bii 288 . . . . . . 7  |-  ( y  e.  A  <->  ( A  \  { y } ) 
C.  A )
149, 13sylib 188 . . . . . 6  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  C.  A )
15 infdifsn 7357 . . . . . . 7  |-  ( om  ~<_  A  ->  ( A  \  { y } ) 
~~  A )
1615adantr 451 . . . . . 6  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  ~~  A )
1714, 16jca 518 . . . . 5  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  (
( A  \  {
y } )  C.  A  /\  ( A  \  { y } ) 
~~  A ) )
18 psseq1 3263 . . . . . . 7  |-  ( x  =  ( A  \  { y } )  ->  ( x  C.  A 
<->  ( A  \  {
y } )  C.  A ) )
19 breq1 4026 . . . . . . 7  |-  ( x  =  ( A  \  { y } )  ->  ( x  ~~  A 
<->  ( A  \  {
y } )  ~~  A ) )
2018, 19anbi12d 691 . . . . . 6  |-  ( x  =  ( A  \  { y } )  ->  ( ( x 
C.  A  /\  x  ~~  A )  <->  ( ( A  \  { y } )  C.  A  /\  ( A  \  { y } )  ~~  A
) ) )
2120spcegv 2869 . . . . 5  |-  ( ( A  \  { y } )  e.  _V  ->  ( ( ( A 
\  { y } )  C.  A  /\  ( A  \  { y } )  ~~  A
)  ->  E. x
( x  C.  A  /\  x  ~~  A ) ) )
228, 17, 21sylc 56 . . . 4  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
2322ex 423 . . 3  |-  ( om  ~<_  A  ->  ( y  e.  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) ) )
2423exlimdv 1664 . 2  |-  ( om  ~<_  A  ->  ( E. y  y  e.  A  ->  E. x ( x 
C.  A  /\  x  ~~  A ) ) )
253, 24mpd 14 1  |-  ( om  ~<_  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C. wpss 3153   (/)c0 3455   {csn 3640   class class class wbr 4023   omcom 4656    ~~ cen 6860    ~<_ cdom 6861
This theorem is referenced by:  isfin4-2  7940
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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