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Theorem infnninfOLD 7070
Description: Obsolete version of infnninf 7069 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
infnninfOLD (ω × {1o}) ∈ ℕ

Proof of Theorem infnninfOLD
Dummy variables 𝑓 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1lt2o 6391 . . . 4 1o ∈ 2o
21fconst6 5371 . . 3 (ω × {1o}):ω⟶2o
3 2onn 6470 . . . . 5 2o ∈ ω
43elexi 2724 . . . 4 2o ∈ V
5 omex 4554 . . . 4 ω ∈ V
64, 5elmap 6624 . . 3 ((ω × {1o}) ∈ (2o𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
72, 6mpbir 145 . 2 (ω × {1o}) ∈ (2o𝑚 ω)
8 peano2 4556 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9 1oex 6373 . . . . . . 7 1o ∈ V
109fvconst2 5685 . . . . . 6 (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
118, 10syl 14 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
129fvconst2 5685 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
1311, 12eqtr4d 2193 . . . 4 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14 eqimss 3182 . . . 4 (((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖) → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
1513, 14syl 14 . . 3 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
1615rgen 2510 . 2 𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17 fveq1 5469 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18 fveq1 5469 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓𝑖) = ((ω × {1o})‘𝑖))
1917, 18sseq12d 3159 . . . 4 (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
2019ralbidv 2457 . . 3 (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21 df-nninf 7066 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
2220, 21elrab2 2871 . 2 ((ω × {1o}) ∈ ℕ ↔ ((ω × {1o}) ∈ (2o𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
237, 16, 22mpbir2an 927 1 (ω × {1o}) ∈ ℕ
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  wral 2435  wss 3102  {csn 3561  suc csuc 4327  ωcom 4551   × cxp 4586  wf 5168  cfv 5172  (class class class)co 5826  1oc1o 6358  2oc2o 6359  𝑚 cmap 6595  xnninf 7065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-id 4255  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1o 6365  df-2o 6366  df-map 6597  df-nninf 7066
This theorem is referenced by:  fxnn0nninf  10346
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