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Theorem inf0 8693
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 8710. (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 3343 . . . 4 𝑥 ∈ V
2 fr0g 7701 . . . 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 7700 . . . 4 (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
5 peano1 7251 . . . 4 ∅ ∈ ω
6 fnfvelrn 6520 . . . 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 710 . . 3 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
83, 7eqeltrri 2836 . 2 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
9 fvelrnb 6406 . . . . 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 6363 . . . . . . . . . 10 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
1211sucid 5965 . . . . . . . . 9 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)
1311sucex 7177 . . . . . . . . . 10 suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
14 eqid 2760 . . . . . . . . . . 11 (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
15 suceq 5951 . . . . . . . . . . 11 (𝑧 = 𝑣 → suc 𝑧 = suc 𝑣)
16 suceq 5951 . . . . . . . . . . 11 (𝑧 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) → suc 𝑧 = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1714, 15, 16frsucmpt2 7705 . . . . . . . . . 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 709 . . . . . . . . 9 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1912, 18syl5eleqr 2846 . . . . . . . 8 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))
20 eleq1 2827 . . . . . . . 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 7247 . . . . . . . . 9 (𝑓 ∈ ω ↔ suc 𝑓 ∈ ω)
23 fnfvelrn 6520 . . . . . . . . . 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 708 . . . . . . . . 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 556 . . . . . 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 6363 . . . . . . 7 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ V
29 eleq2 2828 . . . . . . . 8 (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑧𝑤𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
30 eleq1 2827 . . . . . . . 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 749 . . . . . . 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 3440 . . . . . 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 3165 . . . 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 1871 . 2 𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
37 fndm 6151 . . . . . 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 2835 . . . 4 dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V
41 fnfun 6149 . . . . 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 7299 . . . 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 2828 . . . 4 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑥𝑦𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
46 eleq2 2828 . . . . . 6 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑧𝑦𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
47 eleq2 2828 . . . . . . . 8 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑤𝑦𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
4847anbi2d 742 . . . . . . 7 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧𝑤𝑤𝑦) ↔ (𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
4948exbidv 1999 . . . . . 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 1998 . . . 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 749 . . 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 3440 . 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 710 1 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  wrex 3051  Vcvv 3340  c0 4058  cmpt 4881  dom cdm 5266  ran crn 5267  cres 5268  suc csuc 5886  Fun wfun 6043   Fn wfn 6044  cfv 6049  ωcom 7231  reccrdg 7675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676
This theorem is referenced by:  axinf  8716
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