Theorem nninfisollemne 7107

Description: Lemma for nninfisol 7109. A case where 𝑁 is a successor and 𝑁 and 𝑋 are not equal. (Contributed by Jim Kingdon, 13-Sep-2024.)

Hypotheses

Ref	Expression
nninfisol.x	⊢ (𝜑 → 𝑋 ∈ ℕ∞)
nninfisol.0	⊢ (𝜑 → (𝑋‘𝑁) = ∅)
nninfisol.n	⊢ (𝜑 → 𝑁 ∈ ω)
nninfisollemne.s	⊢ (𝜑 → 𝑁 ≠ ∅)
nninfisollemne.0	⊢ (𝜑 → (𝑋‘∪ 𝑁) = ∅)

Assertion

Ref	Expression
nninfisollemne	⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Distinct variable group:   𝑖,𝑁

Allowed substitution hints:   𝜑(𝑖)   𝑋(𝑖)

Proof of Theorem nninfisollemne

Step	Hyp	Ref	Expression
1		nninfisollemne.0	. . . . 5 ⊢ (𝜑 → (𝑋‘∪ 𝑁) = ∅)
2	1	adantr 274	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) = ∅)
3		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
4	3	fveq1d 5498	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = (𝑋‘∪ 𝑁))
5		eqid 2170	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
6		eleq1 2233	. . . . . . . . . . 11 ⊢ (𝑖 = ∪ 𝑁 → (𝑖 ∈ 𝑁 ↔ ∪ 𝑁 ∈ 𝑁))
7	6	ifbid 3547	. . . . . . . . . 10 ⊢ (𝑖 = ∪ 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(∪ 𝑁 ∈ 𝑁, 1o, ∅))
8		nninfisol.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ω)
9		nnpredcl 4607	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
10	8, 9	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑁 ∈ ω)
11		nninfisollemne.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ≠ ∅)
12		nnpredlt 4608	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∪ 𝑁 ∈ 𝑁)
13	8, 11, 12	syl2anc 409	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑁 ∈ 𝑁)
14	13	iftrued 3533	. . . . . . . . . . 11 ⊢ (𝜑 → if(∪ 𝑁 ∈ 𝑁, 1o, ∅) = 1o)
15		1lt2o 6421	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
16	14, 15	eqeltrdi 2261	. . . . . . . . . 10 ⊢ (𝜑 → if(∪ 𝑁 ∈ 𝑁, 1o, ∅) ∈ 2o)
17	5, 7, 10, 16	fvmptd3 5589	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = if(∪ 𝑁 ∈ 𝑁, 1o, ∅))
18	17, 14	eqtrd 2203	. . . . . . . 8 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = 1o)
19	18	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = 1o)
20	4, 19	eqtr3d 2205	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) = 1o)
21		1n0 6411	. . . . . 6 ⊢ 1o ≠ ∅
22		pm13.181 2422	. . . . . 6 ⊢ (((𝑋‘∪ 𝑁) = 1o ∧ 1o ≠ ∅) → (𝑋‘∪ 𝑁) ≠ ∅)
23	20, 21, 22	sylancl 411	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) ≠ ∅)
24	23	neneqd 2361	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ¬ (𝑋‘∪ 𝑁) = ∅)
25	2, 24	pm2.65da 656	. . 3 ⊢ (𝜑 → ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
26	25	olcd 729	. 2 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ∨ ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋))
27		df-dc 830	. 2 ⊢ (DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ↔ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ∨ ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋))
28	26, 27	sylibr 133	1 ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∅c0 3414  ifcif 3526  ∪ cuni 3796   ↦ cmpt 4050  ωcom 4574  ‘cfv 5198  1_oc1o 6388  2_oc2o 6389  ℕ_∞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-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  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-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  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-iota 5160  df-fun 5200  df-fv 5206  df-1o 6395  df-2o 6396

This theorem is referenced by:  nninfisol  7109