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Theorem infeq5 7518
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 7524.) 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 3272 . . . . 5  |-  ( x 
C.  U. x  <->  ( x  C_ 
U. x  /\  x  =/=  U. x ) )
2 unieq 3959 . . . . . . . . . 10  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 3977 . . . . . . . . . 10  |-  U. (/)  =  (/)
42, 3syl6req 2429 . . . . . . . . 9  |-  ( x  =  (/)  ->  (/)  =  U. x )
5 eqtr 2397 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  (/)  =  U. x )  ->  x  =  U. x )
64, 5mpdan 650 . . . . . . . 8  |-  ( x  =  (/)  ->  x  = 
U. x )
76necon3i 2582 . . . . . . 7  |-  ( x  =/=  U. x  ->  x  =/=  (/) )
87anim1i 552 . . . . . 6  |-  ( ( x  =/=  U. x  /\  x  C_  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
98ancoms 440 . . . . 5  |-  ( ( x  C_  U. x  /\  x  =/=  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
101, 9sylbi 188 . . . 4  |-  ( x 
C.  U. x  ->  (
x  =/=  (/)  /\  x  C_ 
U. x ) )
1110eximi 1582 . . 3  |-  ( E. x  x  C.  U. x  ->  E. x ( x  =/=  (/)  /\  x  C_  U. x ) )
12 eqid 2380 . . . . 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 2380 . . . . 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 2895 . . . . 5  |-  x  e. 
_V
1512, 13, 14, 14inf3lem7 7515 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
1615exlimiv 1641 . . 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 7517 . 2  |-  ( om  e.  _V  ->  E. x  x  C.  U. x )
1917, 18impbii 181 1  |-  ( E. x  x  C.  U. x 
<->  om  e.  _V )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2543   {crab 2646   _Vcvv 2892    i^i cin 3255    C_ wss 3256    C. wpss 3257   (/)c0 3564   U.cuni 3950    e. cmpt 4200   omcom 4778    |` cres 4813   reccrdg 6596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-reg 7486
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-recs 6562  df-rdg 6597
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