Theorem 0nninf 16669

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.

Step	Hyp	Ref	Expression
1		0lt2o 6614	. . . 4 ⊢ ∅ ∈ 2o
2	1	fconst6 5539	. . 3 ⊢ (ω × {∅}):ω⟶2o
3		2onn 6694	. . . . 5 ⊢ 2o ∈ ω
4	3	elexi 2814	. . . 4 ⊢ 2o ∈ V
5		omex 4693	. . . 4 ⊢ ω ∈ V
6	4, 5	elmap 6851	. . 3 ⊢ ((ω × {∅}) ∈ (2o ↑𝑚 ω) ↔ (ω × {∅}):ω⟶2o)
7	2, 6	mpbir 146	. 2 ⊢ (ω × {∅}) ∈ (2o ↑𝑚 ω)
8		peano2 4695	. . . . . 6 ⊢ (𝑖 ∈ ω → suc 𝑖 ∈ ω)
9		0ex 4217	. . . . . . 7 ⊢ ∅ ∈ V
10	9	fvconst2 5873	. . . . . 6 ⊢ (suc 𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅)
11	8, 10	syl 14	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ∅)
12	9	fvconst2 5873	. . . . 5 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘𝑖) = ∅)
13	11, 12	eqtr4d 2266	. . . 4 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖))
14		eqimss 3280	. . . 4 ⊢ (((ω × {∅})‘suc 𝑖) = ((ω × {∅})‘𝑖) → ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))
15	13, 14	syl 14	. . 3 ⊢ (𝑖 ∈ ω → ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖))
16	15	rgen 2584	. 2 ⊢ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)
17		fveq1 5641	. . . . 5 ⊢ (𝑓 = (ω × {∅}) → (𝑓‘suc 𝑖) = ((ω × {∅})‘suc 𝑖))
18		fveq1 5641	. . . . 5 ⊢ (𝑓 = (ω × {∅}) → (𝑓‘𝑖) = ((ω × {∅})‘𝑖))
19	17, 18	sseq12d 3257	. . . 4 ⊢ (𝑓 = (ω × {∅}) → ((𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)))
20	19	ralbidv 2531	. . 3 ⊢ (𝑓 = (ω × {∅}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖) ↔ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)))
21		df-nninf 7324	. . 3 ⊢ ℕ∞ = {𝑓 ∈ (2o ↑𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓‘𝑖)}
22	20, 21	elrab2 2964	. 2 ⊢ ((ω × {∅}) ∈ ℕ∞ ↔ ((ω × {∅}) ∈ (2o ↑𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {∅})‘suc 𝑖) ⊆ ((ω × {∅})‘𝑖)))
23	7, 16, 22	mpbir2an 950	1 ⊢ (ω × {∅}) ∈ ℕ∞

Syntax hints:   = wceq 1397   ∈ wcel 2201  ∀wral 2509   ⊆ wss 3199  ∅c0 3493  {csn 3670  suc csuc 4464  ωcom 4690   × cxp 4725  ⟶wf 5324  ‘cfv 5328  (class class class)co 6023  2_oc2o 6581   ↑_𝑚 cmap 6822  ℕ_∞xnninf 7323

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1o 6587  df-2o 6588  df-map 6824  df-nninf 7324

This theorem is referenced by:  exmidsbthrlem  16689