ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  infnninfOLD GIF version

Theorem infnninfOLD 7136
Description: Obsolete version of infnninf 7135 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 6456 . . . 4 1o ∈ 2o
21fconst6 5427 . . 3 (ω × {1o}):ω⟶2o
3 2onn 6535 . . . . 5 2o ∈ ω
43elexi 2761 . . . 4 2o ∈ V
5 omex 4604 . . . 4 ω ∈ V
64, 5elmap 6690 . . 3 ((ω × {1o}) ∈ (2o𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
72, 6mpbir 146 . 2 (ω × {1o}) ∈ (2o𝑚 ω)
8 peano2 4606 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9 1oex 6438 . . . . . . 7 1o ∈ V
109fvconst2 5745 . . . . . 6 (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
118, 10syl 14 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
129fvconst2 5745 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
1311, 12eqtr4d 2223 . . . 4 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14 eqimss 3221 . . . 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 2540 . 2 𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17 fveq1 5526 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18 fveq1 5526 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓𝑖) = ((ω × {1o})‘𝑖))
1917, 18sseq12d 3198 . . . 4 (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
2019ralbidv 2487 . . 3 (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21 df-nninf 7132 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
2220, 21elrab2 2908 . 2 ((ω × {1o}) ∈ ℕ ↔ ((ω × {1o}) ∈ (2o𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
237, 16, 22mpbir2an 943 1 (ω × {1o}) ∈ ℕ
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wcel 2158  wral 2465  wss 3141  {csn 3604  suc csuc 4377  ωcom 4601   × cxp 4636  wf 5224  cfv 5228  (class class class)co 5888  1oc1o 6423  2oc2o 6424  𝑚 cmap 6661  xnninf 7131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1o 6430  df-2o 6431  df-map 6663  df-nninf 7132
This theorem is referenced by:  fxnn0nninf  10451
  Copyright terms: Public domain W3C validator