Theorem infnninfOLD 7471

Description: Obsolete version of infnninf 7470 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.

Step	Hyp	Ref	Expression
1		1lt2o 6719	. . . 4 ⊢ 1o ∈ 2o
2	1	fconst6 5595	. . 3 ⊢ (ω × {1o}):ω⟶2o
3		2onn 6798	. . . . 5 ⊢ 2o ∈ ω
4	3	elexi 2834	. . . 4 ⊢ 2o ∈ V
5		omex 4742	. . . 4 ⊢ ω ∈ V
6	4, 5	elmap 6962	. . 3 ⊢ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
7	2, 6	mpbir 146	. 2 ⊢ (ω × {1o}) ∈ (2o ↑𝑚 ω)
8		peano2 4744	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9		1oex 6699	. . . . . . 7 ⊢ 1o ∈ V
10	9	fvconst2 5934	. . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
11	8, 10	syl 14	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
12	9	fvconst2 5934	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
13	11, 12	eqtr4d 2274	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
14		eqimss 3302	. . . 4 ⊢ (((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖) → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
15	13, 14	syl 14	. . 3 ⊢ (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
16	15	rgen 2603	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
17		fveq1 5697	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
18		fveq1 5697	. . . . 5 ⊢ (𝑓 = (ω × {1o}) → (𝑓‘𝑖) = ((ω × {1o})‘𝑖))
19	17, 18	sseq12d 3279	. . . 4 ⊢ (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
20	19	ralbidv 2550	. . 3 ⊢ (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
21		df-nninf 7466	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
22	20, 21	elrab2 2985	. 2 ⊢ ((ω × {1o}) ∈ ℕ∞ ↔ ((ω × {1o}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23	7, 16, 22	mpbir2an 955	1 ⊢ (ω × {1o}) ∈ ℕ∞

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  {csn 3709  suc csuc 4512  ωcom 4739   × cxp 4774  ⟶wf 5376  ‘cfv 5380  (class class class)co 6088  1_oc1o 6684  2_oc2o 6685   ↑_𝑚 cmap 6926  ℕ_∞xnninf 7465

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1o 6691  df-2o 6692  df-map 6928  df-nninf 7466

This theorem is used by:  fxnn0nninf  10897