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Theorem 0nninf 13283
 Description: The zero element of ℕ∞ (the constant sequence equal to ∅). (Contributed by Jim Kingdon, 14-Jul-2022.)
Assertion
Ref Expression
0nninf (ω × {∅}) ∈ ℕ

Proof of Theorem 0nninf
Dummy variables 𝑓 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0lt2o 6338 . . . 4 ∅ ∈ 2o
21fconst6 5322 . . 3 (ω × {∅}):ω⟶2o
3 2onn 6417 . . . . 5 2o ∈ ω
43elexi 2698 . . . 4 2o ∈ V
5 omex 4507 . . . 4 ω ∈ V
64, 5elmap 6571 . . 3 ((ω × {∅}) ∈ (2o𝑚 ω) ↔ (ω × {∅}):ω⟶2o)
72, 6mpbir 145 . 2 (ω × {∅}) ∈ (2o𝑚 ω)
8 peano2 4509 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9 0ex 4055 . . . . . . 7 ∅ ∈ V
109fvconst2 5636 . . . . . 6 (suc 𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅)
118, 10syl 14 . . . . 5 (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅)
129fvconst2 5636 . . . . 5 (𝑖 ∈ ω → ((ω × {∅})‘𝑖) = ∅)
1311, 12eqtr4d 2175 . . . 4 (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖))
14 eqimss 3151 . . . 4 (((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖) → ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))
1513, 14syl 14 . . 3 (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))
1615rgen 2485 . 2 𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)
17 fveq1 5420 . . . . 5 (𝑓 = (ω × {∅}) → (𝑓‘suc 𝑖) = ((ω × {∅})‘suc 𝑖))
18 fveq1 5420 . . . . 5 (𝑓 = (ω × {∅}) → (𝑓𝑖) = ((ω × {∅})‘𝑖))
1917, 18sseq12d 3128 . . . 4 (𝑓 = (ω × {∅}) → ((𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)))
2019ralbidv 2437 . . 3 (𝑓 = (ω × {∅}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)))
21 df-nninf 7007 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
2220, 21elrab2 2843 . 2 ((ω × {∅}) ∈ ℕ ↔ ((ω × {∅}) ∈ (2o𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)))
237, 16, 22mpbir2an 926 1 (ω × {∅}) ∈ ℕ
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ⊆ wss 3071  ∅c0 3363  {csn 3527  suc csuc 4287  ωcom 4504   × cxp 4537  ⟶wf 5119  ‘cfv 5123  (class class class)co 5774  2oc2o 6307   ↑𝑚 cmap 6542  ℕ∞xnninf 7005 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1o 6313  df-2o 6314  df-map 6544  df-nninf 7007 This theorem is referenced by:  exmidsbthrlem  13303
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