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Theorem infpss 8097
Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 8193. (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 7369 . . 3  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2 n0 3637 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
31, 2sylib 189 . 2  |-  ( om  ~<_  A  ->  E. y 
y  e.  A )
4 reldom 7115 . . . . . 6  |-  Rel  ~<_
54brrelex2i 4919 . . . . 5  |-  ( om  ~<_  A  ->  A  e.  _V )
6 difexg 4351 . . . . 5  |-  ( A  e.  _V  ->  ( A  \  { y } )  e.  _V )
75, 6syl 16 . . . 4  |-  ( om  ~<_  A  ->  ( A  \  { y } )  e.  _V )
87adantr 452 . . 3  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  e.  _V )
9 simpr 448 . . . . 5  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  y  e.  A )
10 difsnpss 3941 . . . . 5  |-  ( y  e.  A  <->  ( A  \  { y } ) 
C.  A )
119, 10sylib 189 . . . 4  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  C.  A )
12 infdifsn 7611 . . . . 5  |-  ( om  ~<_  A  ->  ( A  \  { y } ) 
~~  A )
1312adantr 452 . . . 4  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  ~~  A )
1411, 13jca 519 . . 3  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  (
( A  \  {
y } )  C.  A  /\  ( A  \  { y } ) 
~~  A ) )
15 psseq1 3434 . . . . 5  |-  ( x  =  ( A  \  { y } )  ->  ( x  C.  A 
<->  ( A  \  {
y } )  C.  A ) )
16 breq1 4215 . . . . 5  |-  ( x  =  ( A  \  { y } )  ->  ( x  ~~  A 
<->  ( A  \  {
y } )  ~~  A ) )
1715, 16anbi12d 692 . . . 4  |-  ( x  =  ( A  \  { y } )  ->  ( ( x 
C.  A  /\  x  ~~  A )  <->  ( ( A  \  { y } )  C.  A  /\  ( A  \  { y } )  ~~  A
) ) )
1817spcegv 3037 . . 3  |-  ( ( A  \  { y } )  e.  _V  ->  ( ( ( A 
\  { y } )  C.  A  /\  ( A  \  { y } )  ~~  A
)  ->  E. x
( x  C.  A  /\  x  ~~  A ) ) )
198, 14, 18sylc 58 . 2  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
203, 19exlimddv 1648 1  |-  ( om  ~<_  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    C. wpss 3321   (/)c0 3628   {csn 3814   class class class wbr 4212   omcom 4845    ~~ cen 7106    ~<_ cdom 7107
This theorem is referenced by:  isfin4-2  8194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113
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