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Theorem infpss 7859
Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 7955. (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 7135 . . 3  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2 n0 3477 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
31, 2sylib 188 . 2  |-  ( om  ~<_  A  ->  E. y 
y  e.  A )
4 reldom 6885 . . . . . . . 8  |-  Rel  ~<_
54brrelex2i 4746 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
6 difexg 4178 . . . . . . 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 3706 . . . . . . . . 9  |-  ( ( A  i^i  { y } )  =  (/)  <->  -.  y  e.  A )
11 disj4 3516 . . . . . . . . 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 7373 . . . . . . 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 3276 . . . . . . 7  |-  ( x  =  ( A  \  { y } )  ->  ( x  C.  A 
<->  ( A  \  {
y } )  C.  A ) )
19 breq1 4042 . . . . . . 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 2882 . . . . 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 1626 . 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 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C. wpss 3166   (/)c0 3468   {csn 3653   class class class wbr 4039   omcom 4672    ~~ cen 6876    ~<_ cdom 6877
This theorem is referenced by:  isfin4-2  7956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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