Theorem nninfisollemne 7329

Description: Lemma for nninfisol 7331. 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 5641	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = (𝑋‘∪ 𝑁))
5		eqid 2231	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
6		eleq1 2294	. . . . . . . . . . 11 ⊢ (𝑖 = ∪ 𝑁 → (𝑖 ∈ 𝑁 ↔ ∪ 𝑁 ∈ 𝑁))
7	6	ifbid 3627	. . . . . . . . . 10 ⊢ (𝑖 = ∪ 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(∪ 𝑁 ∈ 𝑁, 1o, ∅))
8		nninfisol.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ω)
9		nnpredcl 4721	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
10	8, 9	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑁 ∈ ω)
11		nninfisollemne.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ≠ ∅)
12		nnpredlt 4722	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∪ 𝑁 ∈ 𝑁)
13	8, 11, 12	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ 𝑁 ∈ 𝑁)
14	13	iftrued 3612	. . . . . . . . . . 11 ⊢ (𝜑 → if(∪ 𝑁 ∈ 𝑁, 1o, ∅) = 1o)
15		1lt2o 6609	. . . . . . . . . . 11 ⊢ 1o ∈ 2o
16	14, 15	eqeltrdi 2322	. . . . . . . . . 10 ⊢ (𝜑 → if(∪ 𝑁 ∈ 𝑁, 1o, ∅) ∈ 2o)
17	5, 7, 10, 16	fvmptd3 5740	. . . . . . . . 9 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘∪ 𝑁) = if(∪ 𝑁 ∈ 𝑁, 1o, ∅))
18	17, 14	eqtrd 2264	. . . . . . . 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 2266	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘∪ 𝑁) = 1o)
21		1n0 6599	. . . . . 6 ⊢ 1o ≠ ∅
22		pm13.181 2484	. . . . . 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 2423	. . . 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 741	. 2 ⊢ (𝜑 → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ∨ ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋))
27		df-dc 842	. 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 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∅c0 3494  ifcif 3605  ∪ cuni 3893   ↦ cmpt 4150  ωcom 4688  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575  ℕ_∞xnninf 7317

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-1o 6581  df-2o 6582

This theorem is referenced by:  nninfisol  7331