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Theorem infpss 7838
Description: Every infinite set has an equinumerous proper subset. Exercise 7 of [TakeutiZaring] p. 91. (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
StepHypRef Expression
1 infn0 7114 . . 3  |-  ( om  ~<_  A  ->  A  =/=  (/) )
2 n0 3465 . . 3  |-  ( A  =/=  (/)  <->  E. y  y  e.  A )
31, 2sylib 190 . 2  |-  ( om  ~<_  A  ->  E. y 
y  e.  A )
4 reldom 6864 . . . . . . . 8  |-  Rel  ~<_
54brrelex2i 4729 . . . . . . 7  |-  ( om  ~<_  A  ->  A  e.  _V )
6 difexg 4163 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  \  { y } )  e.  _V )
75, 6syl 17 . . . . . 6  |-  ( om  ~<_  A  ->  ( A  \  { y } )  e.  _V )
87adantr 453 . . . . 5  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  e.  _V )
9 simpr 449 . . . . . . 7  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  y  e.  A )
10 disjsn 3694 . . . . . . . . 9  |-  ( ( A  i^i  { y } )  =  (/)  <->  -.  y  e.  A )
11 disj4 3504 . . . . . . . . 9  |-  ( ( A  i^i  { y } )  =  (/)  <->  -.  ( A  \  { y } )  C.  A
)
1210, 11bitr3i 244 . . . . . . . 8  |-  ( -.  y  e.  A  <->  -.  ( A  \  { y } )  C.  A )
1312con4bii 290 . . . . . . 7  |-  ( y  e.  A  <->  ( A  \  { y } ) 
C.  A )
149, 13sylib 190 . . . . . 6  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  C.  A )
15 infdifsn 7352 . . . . . . 7  |-  ( om  ~<_  A  ->  ( A  \  { y } ) 
~~  A )
1615adantr 453 . . . . . 6  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  ( A  \  { y } )  ~~  A )
1714, 16jca 520 . . . . 5  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  (
( A  \  {
y } )  C.  A  /\  ( A  \  { y } ) 
~~  A ) )
18 psseq1 3264 . . . . . . 7  |-  ( x  =  ( A  \  { y } )  ->  ( x  C.  A 
<->  ( A  \  {
y } )  C.  A ) )
19 breq1 4027 . . . . . . 7  |-  ( x  =  ( A  \  { y } )  ->  ( x  ~~  A 
<->  ( A  \  {
y } )  ~~  A ) )
2018, 19anbi12d 694 . . . . . 6  |-  ( x  =  ( A  \  { y } )  ->  ( ( x 
C.  A  /\  x  ~~  A )  <->  ( ( A  \  { y } )  C.  A  /\  ( A  \  { y } )  ~~  A
) ) )
2120spcegv 2870 . . . . 5  |-  ( ( A  \  { y } )  e.  _V  ->  ( ( ( A 
\  { y } )  C.  A  /\  ( A  \  { y } )  ~~  A
)  ->  E. x
( x  C.  A  /\  x  ~~  A ) ) )
228, 17, 21sylc 58 . . . 4  |-  ( ( om  ~<_  A  /\  y  e.  A )  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
2322ex 425 . . 3  |-  ( om  ~<_  A  ->  ( y  e.  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) ) )
2423exlimdv 1668 . 2  |-  ( om  ~<_  A  ->  ( E. y  y  e.  A  ->  E. x ( x 
C.  A  /\  x  ~~  A ) ) )
253, 24mpd 16 1  |-  ( om  ~<_  A  ->  E. x
( x  C.  A  /\  x  ~~  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360   E.wex 1533    = wceq 1628    e. wcel 1688    =/= wne 2447   _Vcvv 2789    \ cdif 3150    i^i cin 3152    C. wpss 3154   (/)c0 3456   {csn 3641   class class class wbr 4024   omcom 4655    ~~ cen 6855    ~<_ cdom 6856
This theorem is referenced by:  isfin4-2  7935
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-1o 6474  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862
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