Theorem nninfisollemne 7435

Description: Lemma for nninfisol 7437. 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 276	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) = ∅)
3		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
4	3	fveq1d 5677	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = (𝑋‘∪ 𝑁))
5		eqid 2234	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
6		eleq1 2297	. . . . . . . . . . 11 ⊢ (𝑖 = ∪ 𝑁 → (𝑖 ∈ 𝑁 ↔ ∪ 𝑁 ∈ 𝑁))
7	6	ifbid 3648	. . . . . . . . . 10 ⊢ (𝑖 = ∪ 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(∪ 𝑁 ∈ 𝑁, 1o, ∅))
8		nninfisol.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ω)
9		nnpredcl 4750	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
10	8, 9	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑁 ∈ ω)
11		nninfisollemne.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ≠ ∅)
12		nnpredlt 4751	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∪ 𝑁 ∈ 𝑁)
13	8, 11, 12	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑁 ∈ 𝑁)
14	13	iftrued 3633	. . . . . . . . . . 11 ⊢ (𝜑 → if(∪ 𝑁 ∈ 𝑁, 1o, ∅) = 1o)
15		1lt2o 6688	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
16	14, 15	eqeltrdi 2325	. . . . . . . . . 10 ⊢ (𝜑 → if(∪ 𝑁 ∈ 𝑁, 1o, ∅) ∈ 2o)
17	5, 7, 10, 16	fvmptd3 5776	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = if(∪ 𝑁 ∈ 𝑁, 1o, ∅))
18	17, 14	eqtrd 2267	. . . . . . . 8 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = 1o)
19	18	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = 1o)
20	4, 19	eqtr3d 2269	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) = 1o)
21		1n0 6678	. . . . . 6 ⊢ 1o ≠ ∅
22		pm13.181 2496	. . . . . 6 ⊢ (((𝑋‘∪ 𝑁) = 1o ∧ 1o ≠ ∅) → (𝑋‘∪ 𝑁) ≠ ∅)
23	20, 21, 22	sylancl 413	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) ≠ ∅)
24	23	neneqd 2435	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ¬ (𝑋‘∪ 𝑁) = ∅)
25	2, 24	pm2.65da 667	. . 3 ⊢ (𝜑 → ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
26	25	olcd 742	. 2 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ∨ ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋))
27		df-dc 843	. 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 134	1 ⊢ (𝜑 → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  ∅c0 3512  ifcif 3624  ∪ cuni 3919   ↦ cmpt 4176  ωcom 4717  ‘cfv 5357  1_oc1o 6653  2_oc2o 6654  ℕ_∞xnninf 7423

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-1o 6660  df-2o 6661

This theorem is referenced by:  nninfisol  7437