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Theorem inf0 9068
 Description: Existence of ω implies our axiom of infinity ax-inf 9085. 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 9085. (Contributed by NM, 15-Oct-1996.) Revised to closed form. (Revised by BJ, 20-May-2024.)
Assertion
Ref Expression
inf0 (ω ∈ 𝑉 → ∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem inf0
Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fr0g 8054 . . . 4 (𝑥 ∈ V → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥)
21elv 3484 . . 3 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) = 𝑥
3 frfnom 8053 . . . 4 (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω
4 peano1 7584 . . . 4 ∅ ∈ ω
5 fnfvelrn 6829 . . . 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))
63, 4, 5mp2an 691 . . 3 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘∅) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
72, 6eqeltrri 2913 . 2 𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
8 fvelrnb 6707 . . . . 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))
93, 8ax-mp 5 . . . 4 (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ↔ ∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧)
10 fvex 6664 . . . . . . . . . 10 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
1110sucid 6251 . . . . . . . . 9 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓)
1210sucex 7509 . . . . . . . . . 10 suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ V
13 eqid 2824 . . . . . . . . . . 11 (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
14 suceq 6237 . . . . . . . . . . 11 (𝑧 = 𝑣 → suc 𝑧 = suc 𝑣)
15 suceq 6237 . . . . . . . . . . 11 (𝑧 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) → suc 𝑧 = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1613, 14, 15frsucmpt2 8059 . . . . . . . . . 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))
1712, 16mpan2 690 . . . . . . . . 9 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) = suc ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓))
1811, 17eleqtrrid 2923 . . . . . . . 8 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓))
19 eleq1 2903 . . . . . . . 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)))
2018, 19syl5ib 247 . . . . . . 7 (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → (𝑓 ∈ ω → 𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
21 peano2b 7579 . . . . . . . 8 (𝑓 ∈ ω ↔ suc 𝑓 ∈ ω)
22 fnfvelrn 6829 . . . . . . . . 9 (((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))
233, 22mpan 689 . . . . . . . 8 (suc 𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
2421, 23sylbi 220 . . . . . . 7 (𝑓 ∈ ω → ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
2520, 24jca2 517 . . . . . 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))))
26 fvex 6664 . . . . . . 7 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) ∈ V
27 eleq2 2904 . . . . . . . 8 (𝑤 = ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓) → (𝑧𝑤𝑧 ∈ ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘suc 𝑓)))
28 eleq1 2903 . . . . . . . 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)))
2927, 28anbi12d 633 . . . . . . 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))))
3026, 29spcev 3592 . . . . . 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)))
3125, 30syl6com 37 . . . . 5 (𝑓 ∈ ω → (((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
3231rexlimiv 3272 . . . 4 (∃𝑓 ∈ ω ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)‘𝑓) = 𝑧 → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
339, 32sylbi 220 . . 3 (𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
3433ax-gen 1797 . 2 𝑧(𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
35 fndm 6436 . . . . . 6 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = ω)
363, 35ax-mp 5 . . . . 5 dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) = ω
37 id 22 . . . . 5 (ω ∈ 𝑉 → ω ∈ 𝑉)
3836, 37eqeltrid 2920 . . . 4 (ω ∈ 𝑉 → dom (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ 𝑉)
39 fnfun 6434 . . . . 5 ((rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) Fn ω → Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))
403, 39ax-mp 5 . . . 4 Fun (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)
41 funrnex 7638 . . . 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))
4238, 40, 41mpisyl 21 . . 3 (ω ∈ 𝑉 → ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V)
43 eleq2 2904 . . . . 5 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑥𝑦𝑥 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
44 eleq2 2904 . . . . . . 7 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑧𝑦𝑧 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
45 eleq2 2904 . . . . . . . . 9 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (𝑤𝑦𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω)))
4645anbi2d 631 . . . . . . . 8 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → ((𝑧𝑤𝑤𝑦) ↔ (𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
4746exbidv 1923 . . . . . . 7 (𝑦 = ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) → (∃𝑤(𝑧𝑤𝑤𝑦) ↔ ∃𝑤(𝑧𝑤𝑤 ∈ ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω))))
4844, 47imbi12d 348 . . . . . 6 (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)))))
4948albidv 1922 . . . . 5 (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)))))
5043, 49anbi12d 633 . . . 4 (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))))))
5150spcegv 3582 . . 3 (ran (rec((𝑣 ∈ V ↦ suc 𝑣), 𝑥) ↾ ω) ∈ V → ((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)))))
5242, 51syl 17 . 2 (ω ∈ 𝑉 → ((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)))))
537, 34, 52mp2ani 697 1 (ω ∈ 𝑉 → ∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2115  ∃wrex 3133  Vcvv 3479  ∅c0 4274   ↦ cmpt 5127  dom cdm 5536  ran crn 5537   ↾ cres 5538  suc csuc 6174  Fun wfun 6330   Fn wfn 6331  ‘cfv 6336  ωcom 7563  reccrdg 8028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7564  df-wrecs 7930  df-recs 7991  df-rdg 8029 This theorem is referenced by:  axinf  9091
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