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Theorem infeq5 7584
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 7590.) 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 3328 . . . . 5  |-  ( x 
C.  U. x  <->  ( x  C_ 
U. x  /\  x  =/=  U. x ) )
2 unieq 4016 . . . . . . . . . 10  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 4034 . . . . . . . . . 10  |-  U. (/)  =  (/)
42, 3syl6req 2484 . . . . . . . . 9  |-  ( x  =  (/)  ->  (/)  =  U. x )
5 eqtr 2452 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  (/)  =  U. x )  ->  x  =  U. x )
64, 5mpdan 650 . . . . . . . 8  |-  ( x  =  (/)  ->  x  = 
U. x )
76necon3i 2637 . . . . . . 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 1585 . . 3  |-  ( E. x  x  C.  U. x  ->  E. x ( x  =/=  (/)  /\  x  C_  U. x ) )
12 eqid 2435 . . . . 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 2435 . . . . 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 2951 . . . . 5  |-  x  e. 
_V
1512, 13, 14, 14inf3lem7 7581 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
1615exlimiv 1644 . . 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 7583 . 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 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   {crab 2701   _Vcvv 2948    i^i cin 3311    C_ wss 3312    C. wpss 3313   (/)c0 3620   U.cuni 4007    e. cmpt 4258   omcom 4837    |` cres 4872   reccrdg 6659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-reg 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
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