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Theorem inf3 7336
Description: Our Axiom of Infinity ax-inf 7339 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 7324, and the conclusion is the version of the Axiom of Infinity shown as Axiom 7 in [TakeutiZaring] p. 43. (Other standard versions are proved later as axinf2 7341 and zfinf2 7343.) The main proof is provided by inf3lema 7325 through inf3lem7 7335, and this final piece eliminates the auxiliary hypothesis of inf3lem7 7335. This proof is due to Ian Sutherland, Richard Heck, and Norman Megill and was posted on Usenet as shown below. Although the result is not new, the authors were unable to find a published proof.
       (As posted to sci.logic on 30-Oct-1996, with annotations added.)

       Theorem:  The statement "There exists a non-empty set that is a subset
       of its union" implies the Axiom of Infinity.

       Proof:  Let X be a nonempty set which is a subset of its union; the
       latter
       property is equivalent to saying that for any y in X, there exists a z
       in X
       such that y is in z.

       Define by finite recursion a function F:omega-->(power X) such that
       F_0 = 0  (See inf3lemb 7326.)
       F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 7327.)
       Note: ^ means intersect, < means \in ("element of").
       (Finite recursion as typically done requires the existence of omega;
       to avoid this we can just use transfinite recursion restricted to omega.
       F is a class-term that is not necessarily a set at this point.)

       Lemma 1.  F_n subset F_n+1.  (See inf3lem1 7329.)
       Proof:  By induction:  F_0 subset F_1.  If y < F_n+1, then y^X subset
       F_n,
       so if F_n subset F_n+1, then y^X subset F_n+1, so y < F_n+2.

       Lemma 2.  F_n =/= X.  (See inf3lem2 7330.)
       Proof:  By induction:  F_0 =/= X because X is not empty.  Assume F_n =/=
       X.
       Then there is a y in X that is not in F_n.  By definition of X, there is
       a
       z in X that contains y.  Suppose F_n+1 = X.  Then z is in F_n+1, and z^X
       contains y, so z^X is not a subset of F_n, contrary to the definition of
       F_n+1.

       Lemma 3.  F_n =/= F_n+1.  (See inf3lem3 7331.)
       Proof:  Using the identity y^X subset F_n <-> y^(X-F_n) = 0, we have
       F_n+1 = {y<X | y^(X-F_n) = 0}.  Let q = {y<X-F_n | y^(X-F_n) = 0}.
       Then q subset F_n+1.  Since X-F_n is not empty by Lemma 2 and q is the
       set of \in-minimal elements of X-F_n, by Foundation q is not empty, so q
       and therefore F_n+1 have an element not in F_n.

       Lemma 4.  F_n proper_subset F_n+1.  (See inf3lem4 7332.)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 7333.)
       Proof:  Fix m and use induction on n > m.  Basis: F_m proper_subset
       F_m+1
       by Lemma 4.  Induction:  Assume F_m proper_subset F_n.  Then since F_n
       proper_subset F_n+1, F_m proper_subset F_n+1 by transitivity of proper
       subset.

       By Lemma 5, F_m =/= F_n for m =/= n, so F is 1-1.  (See inf3lem6 7334.)
       Thus the inverse of F is a function with range omega and domain a subset
       of power X, so omega exists by Replacement.  (See inf3lem7 7335.)
       Q.E.D.
       
(Contributed by NM, 29-Oct-1996.)
Hypothesis
Ref Expression
inf3.1  |-  E. x
( x  =/=  (/)  /\  x  C_ 
U. x )
Assertion
Ref Expression
inf3  |-  om  e.  _V

Proof of Theorem inf3
Dummy variables  y  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inf3.1 . 2  |-  E. x
( x  =/=  (/)  /\  x  C_ 
U. x )
2 eqid 2283 . . . 4  |-  ( y  e.  _V  |->  { w  e.  x  |  (
w  i^i  x )  C_  y } )  =  ( y  e.  _V  |->  { w  e.  x  |  ( w  i^i  x )  C_  y } )
3 eqid 2283 . . . 4  |-  ( 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 )
4 vex 2791 . . . 4  |-  x  e. 
_V
52, 3, 4, 4inf3lem7 7335 . . 3  |-  ( ( x  =/=  (/)  /\  x  C_ 
U. x )  ->  om  e.  _V )
65exlimiv 1666 . 2  |-  ( E. x ( x  =/=  (/)  /\  x  C_  U. x
)  ->  om  e.  _V )
71, 6ax-mp 8 1  |-  om  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827    e. cmpt 4077   omcom 4656    |` cres 4691   reccrdg 6422
This theorem is referenced by:  axinf2  7341
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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