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Theorem infeq5 7338
Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (shown on the right-hand side in the form of omex 7344.) The left-hand side provides us with a very short way to express of the Axiom of Infinity using only elementary symbols. This proof of equivalence does not depend on the Axiom of Infinity. (Contributed by NM, 23-Mar-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
infeq5  |-  ( E. x  x  C.  U. x 
<->  om  e.  _V )

Proof of Theorem infeq5
Dummy variables  y  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pss 3168 . . . . 5  |-  ( x 
C.  U. x  <->  ( x  C_ 
U. x  /\  x  =/=  U. x ) )
2 unieq 3836 . . . . . . . . . 10  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 3854 . . . . . . . . . 10  |-  U. (/)  =  (/)
42, 3syl6req 2332 . . . . . . . . 9  |-  ( x  =  (/)  ->  (/)  =  U. x )
5 eqtr 2300 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  (/)  =  U. x )  ->  x  =  U. x )
64, 5mpdan 649 . . . . . . . 8  |-  ( x  =  (/)  ->  x  = 
U. x )
76necon3i 2485 . . . . . . 7  |-  ( x  =/=  U. x  ->  x  =/=  (/) )
87anim1i 551 . . . . . 6  |-  ( ( x  =/=  U. x  /\  x  C_  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
98ancoms 439 . . . . 5  |-  ( ( x  C_  U. x  /\  x  =/=  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
101, 9sylbi 187 . . . 4  |-  ( x 
C.  U. x  ->  (
x  =/=  (/)  /\  x  C_ 
U. x ) )
1110eximi 1563 . . 3  |-  ( E. x  x  C.  U. x  ->  E. x ( x  =/=  (/)  /\  x  C_  U. x ) )
12 eqid 2283 . . . . 5  |-  ( y  e.  _V  |->  { w  e.  x  |  (
w  i^i  x )  C_  y } )  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
13 eqid 2283 . . . . 5  |-  ( rec ( ( y  e. 
_V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } ) ,  (/) )  |`  om )  =  ( rec ( ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x
)  C_  y }
) ,  (/) )  |`  om )
14 vex 2791 . . . . 5  |-  x  e. 
_V
1512, 13, 14, 14inf3lem7 7335 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
1615exlimiv 1666 . . 3  |-  ( E. x ( x  =/=  (/)  /\  x  C_  U. x
)  ->  om  e.  _V )
1711, 16syl 15 . 2  |-  ( E. x  x  C.  U. x  ->  om  e.  _V )
18 infeq5i 7337 . 2  |-  ( om  e.  _V  ->  E. x  x  C.  U. x )
1917, 18impbii 180 1  |-  ( E. x  x  C.  U. x 
<->  om  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152    C. wpss 3153   (/)c0 3455   U.cuni 3827    e. cmpt 4077   omcom 4656    |` cres 4691   reccrdg 6422
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306
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-reu 2550  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-iun 3907  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-recs 6388  df-rdg 6423
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