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Theorem inf3 8393
Description: Our Axiom of Infinity ax-inf 8396 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 8381, 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 8398 and zfinf2 8400.) The main proof is provided by inf3lema 8382 through inf3lem7 8392, and this final piece eliminates the auxiliary hypothesis of inf3lem7 8392. 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 nonempty 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 8383.)
       F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 8384.)
       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 8386.)
       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 8387.)
       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 8388.)
       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 8389.)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 8390.)
       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 8391.)
       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 8392.)
       Q.E.D.
       
(Contributed by NM, 29-Oct-1996.)
Hypothesis
Ref Expression
inf3.1 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
Assertion
Ref Expression
inf3 ω ∈ V

Proof of Theorem inf3
Dummy variables 𝑦 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2609 . . 3 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
2 eqid 2609 . . 3 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
3 vex 3175 . . 3 𝑥 ∈ V
41, 2, 3, 3inf3lem7 8392 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
5 inf3.1 . 2 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
64, 5exlimiiv 1845 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 382  wex 1694  wcel 1976  wne 2779  {crab 2899  Vcvv 3172  cin 3538  wss 3539  c0 3873   cuni 4366  cmpt 4637  cres 5030  ωcom 6935  reccrdg 7370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-reg 8358
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-om 6936  df-wrecs 7272  df-recs 7333  df-rdg 7371
This theorem is referenced by:  axinf2  8398
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