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Theorem inf3 9630
Description: Our Axiom of Infinity ax-inf 9633 implies the standard Axiom of Infinity. The hypothesis is a variant of our Axiom of Infinity provided by inf2 9618, 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 9635 and zfinf2 9637.) The main proof is provided by inf3lema 9619 through inf3lem7 9629, and this final piece eliminates the auxiliary hypothesis of inf3lem7 9629. 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 9620.)
       F_n+1 = {y<X | y^X subset F_n}  (See inf3lemc 9621.)
       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 9623.)
       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 9624.)
       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 9625.)
       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 9626.)
       Proof:  Lemmas 1 and 3.

       Lemma 5.  F_m proper_subset F_n, m < n.  (See inf3lem5 9627.)
       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 9628.)
       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 9629.)
       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 2733 . . 3 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
2 eqid 2733 . . 3 (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}), ∅) ↾ ω)
3 vex 3479 . . 3 𝑥 ∈ V
41, 2, 3, 3inf3lem7 9629 . 2 ((𝑥 ≠ ∅ ∧ 𝑥 𝑥) → ω ∈ V)
5 inf3.1 . 2 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
64, 5exlimiiv 1935 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 397  wex 1782  wcel 2107  wne 2941  {crab 3433  Vcvv 3475  cin 3948  wss 3949  c0 4323   cuni 4909  cmpt 5232  cres 5679  ωcom 7855  reccrdg 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410
This theorem is referenced by:  axinf2  9635
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