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Theorem infnninfOLD 7101
Description: Obsolete version of infnninf 7100 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 6421 . . . 4 1o ∈ 2o
21fconst6 5397 . . 3 (ω × {1o}):ω⟶2o
3 2onn 6500 . . . . 5 2o ∈ ω
43elexi 2742 . . . 4 2o ∈ V
5 omex 4577 . . . 4 ω ∈ V
64, 5elmap 6655 . . 3 ((ω × {1o}) ∈ (2o𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
72, 6mpbir 145 . 2 (ω × {1o}) ∈ (2o𝑚 ω)
8 peano2 4579 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9 1oex 6403 . . . . . . 7 1o ∈ V
109fvconst2 5712 . . . . . 6 (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
118, 10syl 14 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
129fvconst2 5712 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
1311, 12eqtr4d 2206 . . . 4 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14 eqimss 3201 . . . 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 2523 . 2 𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17 fveq1 5495 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18 fveq1 5495 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓𝑖) = ((ω × {1o})‘𝑖))
1917, 18sseq12d 3178 . . . 4 (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
2019ralbidv 2470 . . 3 (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21 df-nninf 7097 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
2220, 21elrab2 2889 . 2 ((ω × {1o}) ∈ ℕ ↔ ((ω × {1o}) ∈ (2o𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
237, 16, 22mpbir2an 937 1 (ω × {1o}) ∈ ℕ
Colors of variables: wff set class
Syntax hints:   = wceq 1348  wcel 2141  wral 2448  wss 3121  {csn 3583  suc csuc 4350  ωcom 4574   × cxp 4609  wf 5194  cfv 5198  (class class class)co 5853  1oc1o 6388  2oc2o 6389  𝑚 cmap 6626  xnninf 7096
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628  df-nninf 7097
This theorem is referenced by:  fxnn0nninf  10394
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