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Theorem infeq5 8045
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 8051.) 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 3477 . . . . 5  |-  ( x 
C.  U. x  <->  ( x  C_ 
U. x  /\  x  =/=  U. x ) )
2 unieq 4243 . . . . . . . . . 10  |-  ( x  =  (/)  ->  U. x  =  U. (/) )
3 uni0 4262 . . . . . . . . . 10  |-  U. (/)  =  (/)
42, 3syl6req 2512 . . . . . . . . 9  |-  ( x  =  (/)  ->  (/)  =  U. x )
5 eqtr 2480 . . . . . . . . 9  |-  ( ( x  =  (/)  /\  (/)  =  U. x )  ->  x  =  U. x )
64, 5mpdan 666 . . . . . . . 8  |-  ( x  =  (/)  ->  x  = 
U. x )
76necon3i 2694 . . . . . . 7  |-  ( x  =/=  U. x  ->  x  =/=  (/) )
87anim1i 566 . . . . . 6  |-  ( ( x  =/=  U. x  /\  x  C_  U. x
)  ->  ( x  =/=  (/)  /\  x  C_  U. x ) )
98ancoms 451 . . . . 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 1661 . . 3  |-  ( E. x  x  C.  U. x  ->  E. x ( x  =/=  (/)  /\  x  C_  U. x ) )
12 eqid 2454 . . . . 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 2454 . . . . 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 3109 . . . . 5  |-  x  e. 
_V
1512, 13, 14, 14inf3lem7 8042 . . . 4  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
1615exlimiv 1727 . . 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 8044 . 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 367    = wceq 1398   E.wex 1617    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461    C. wpss 3462   (/)c0 3783   U.cuni 4235    |-> cmpt 4497    |` cres 4990   omcom 6673   reccrdg 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-reg 8010
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-recs 7034  df-rdg 7068
This theorem is referenced by: (None)
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