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Theorem infeq5 7946
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 7952.) The left-hand side provides us with a very short way to express 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 3444 . . . . 5  |-  ( x 
C.  U. x  <->  ( x  C_ 
U. x  /\  x  =/=  U. x ) )
2 unieq 4199 . . . . . . . . . 10  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 4218 . . . . . . . . . 10  |-  U. (/)  =  (/)
42, 3syl6req 2509 . . . . . . . . 9  |-  ( x  =  (/)  ->  (/)  =  U. x )
5 eqtr 2477 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  (/)  =  U. x )  ->  x  =  U. x )
64, 5mpdan 668 . . . . . . . 8  |-  ( x  =  (/)  ->  x  = 
U. x )
76necon3i 2688 . . . . . . 7  |-  ( x  =/=  U. x  ->  x  =/=  (/) )
87anim1i 568 . . . . . 6  |-  ( ( x  =/=  U. x  /\  x  C_  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
98ancoms 453 . . . . 5  |-  ( ( x  C_  U. x  /\  x  =/=  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
101, 9sylbi 195 . . . 4  |-  ( x 
C.  U. x  ->  (
x  =/=  (/)  /\  x  C_ 
U. x ) )
1110eximi 1626 . . 3  |-  ( E. x  x  C.  U. x  ->  E. x ( x  =/=  (/)  /\  x  C_  U. x ) )
12 eqid 2451 . . . . 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 2451 . . . . 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 3073 . . . . 5  |-  x  e. 
_V
1512, 13, 14, 14inf3lem7 7943 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
1615exlimiv 1689 . . 3  |-  ( E. x ( x  =/=  (/)  /\  x  C_  U. x
)  ->  om  e.  _V )
1711, 16syl 16 . 2  |-  ( E. x  x  C.  U. x  ->  om  e.  _V )
18 infeq5i 7945 . 2  |-  ( om  e.  _V  ->  E. x  x  C.  U. x )
1917, 18impbii 188 1  |-  ( E. x  x  C.  U. x  <->  om  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   {crab 2799   _Vcvv 3070    i^i cin 3427    C_ wss 3428    C. wpss 3429   (/)c0 3737   U.cuni 4191    |-> cmpt 4450    |` cres 4942   omcom 6578   reccrdg 6967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-reg 7910
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-om 6579  df-recs 6934  df-rdg 6968
This theorem is referenced by: (None)
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