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Theorem inf0 8682
Description: Our Axiom of Infinity derived from existence of omega. The proof shows that the especially contrived class "ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) " exists, is a subset of its union, and contains a given set 𝑥 (and thus is nonempty). Thus, it provides an example demonstrating that a set 𝑦 exists with the necessary properties demanded by ax-inf 8699. (Contributed by NM, 15-Oct-1996.)
Hypothesis
Ref Expression
inf0.1 ω ∈ V
Assertion
Ref Expression
inf0 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem inf0
Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3354 . . . 4 𝑥 ∈ V
2 fr0g 7684 . . . 4 (𝑥 ∈ V → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥)
31, 2ax-mp 5 . . 3 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥
4 frfnom 7683 . . . 4 (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
5 peano1 7232 . . . 4 ∅ ∈ ω
6 fnfvelrn 6499 . . . 4 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
74, 5, 6mp2an 672 . . 3 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
83, 7eqeltrri 2847 . 2 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
9 fvelrnb 6385 . . . . 5 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧))
104, 9ax-mp 5 . . . 4 (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧)
11 fvex 6342 . . . . . . . . . 10 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
1211sucid 5947 . . . . . . . . 9 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)
1311sucex 7158 . . . . . . . . . 10 suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
14 eqid 2771 . . . . . . . . . . 11 (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
15 suceq 5933 . . . . . . . . . . 11 (𝑧 = 𝑣 → suc 𝑧 = suc 𝑣)
16 suceq 5933 . . . . . . . . . . 11 (𝑧 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) → suc 𝑧 = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1714, 15, 16frsucmpt2 7688 . . . . . . . . . 10 ((𝑓 ∈ ω ∧ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1813, 17mpan2 671 . . . . . . . . 9 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1912, 18syl5eleqr 2857 . . . . . . . 8 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))
20 eleq1 2838 . . . . . . . 8 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ↔ 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
2119, 20syl5ib 234 . . . . . . 7 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
22 peano2b 7228 . . . . . . . . 9 (𝑓 ∈ ω ↔ suc 𝑓 ∈ ω)
23 fnfvelrn 6499 . . . . . . . . . 10 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω ∧ suc 𝑓 ∈ ω) → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
244, 23mpan 670 . . . . . . . . 9 (suc 𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
2522, 24sylbi 207 . . . . . . . 8 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
2625a1i 11 . . . . . . 7 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
2721, 26jcad 502 . . . . . 6 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → (𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
28 fvex 6342 . . . . . . 7 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ V
29 eleq2 2839 . . . . . . . 8 (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑧𝑤𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
30 eleq1 2838 . . . . . . . 8 (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
3129, 30anbi12d 616 . . . . . . 7 (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → ((𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) ↔ (𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
3228, 31spcev 3451 . . . . . 6 ((𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∧ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
3327, 32syl6com 37 . . . . 5 (𝑓 ∈ ω → (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
3433rexlimiv 3175 . . . 4 (∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
3510, 34sylbi 207 . . 3 (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
3635ax-gen 1870 . 2 𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
37 fndm 6130 . . . . . 6 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = ω)
384, 37ax-mp 5 . . . . 5 dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = ω
39 inf0.1 . . . . 5 ω ∈ V
4038, 39eqeltri 2846 . . . 4 dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V
41 fnfun 6128 . . . . 5 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
424, 41ax-mp 5 . . . 4 Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
43 funrnex 7280 . . . 4 (dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V → (Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V))
4440, 42, 43mp2 9 . . 3 ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V
45 eleq2 2839 . . . 4 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑥𝑦𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
46 eleq2 2839 . . . . . 6 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑧𝑦𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
47 eleq2 2839 . . . . . . . 8 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑤𝑦𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
4847anbi2d 614 . . . . . . 7 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧𝑤𝑤𝑦) ↔ (𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
4948exbidv 2002 . . . . . 6 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∃𝑤(𝑧𝑤𝑤𝑦) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
5046, 49imbi12d 333 . . . . 5 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))))
5150albidv 2001 . . . 4 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))))
5245, 51anbi12d 616 . . 3 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ (𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))))
5344, 52spcev 3451 . 2 ((𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∧ ∀𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))) → ∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))))
548, 36, 53mp2an 672 1 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wex 1852  wcel 2145  wrex 3062  Vcvv 3351  c0 4063  cmpt 4863  dom cdm 5249  ran crn 5250  cres 5251  suc csuc 5868  Fun wfun 6025   Fn wfn 6026  cfv 6031  ωcom 7212  reccrdg 7658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659
This theorem is referenced by:  axinf  8705
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