Theorem peano3nninf 15738

Description: The successor function on ℕ_∞ is never zero. Half of Lemma 3.4 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.)

Hypothesis

Ref	Expression
peano3nninf.s	⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))

Assertion

Ref	Expression
peano3nninf	⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Distinct variable groups:   𝐴,𝑖,𝑝   𝑆,𝑖,𝑥   𝑥,𝑝

Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑝)

Proof of Theorem peano3nninf

Step	Hyp	Ref	Expression
1		fveq1 5560	. . . . . . . . . 10 ⊢ (𝑝 = 𝐴 → (𝑝‘∪ 𝑖) = (𝐴‘∪ 𝑖))
2	1	ifeq2d 3580	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)))
3	2	mpteq2dv 4125	. . . . . . . 8 ⊢ (𝑝 = 𝐴 → (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))))
4		peano3nninf.s	. . . . . . . 8 ⊢ 𝑆 = (𝑝 ∈ ℕ∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖))))
5		omex 4630	. . . . . . . . 9 ⊢ ω ∈ V
6	5	mptex 5791	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))) ∈ V
7	3, 4, 6	fvmpt 5641	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))))
8		eqeq1 2203	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9		unieq 3849	. . . . . . . . . 10 ⊢ (𝑖 = ∅ → ∪ 𝑖 = ∪ ∅)
10	9	fveq2d 5565	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝐴‘∪ 𝑖) = (𝐴‘∪ ∅))
11	8, 10	ifbieq2d 3586	. . . . . . . 8 ⊢ (𝑖 = ∅ → if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
12	11	adantl 277	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ∞ ∧ 𝑖 = ∅) → if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
13		peano1 4631	. . . . . . . 8 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → ∅ ∈ ω)
15		eqid 2196	. . . . . . . . . 10 ⊢ ∅ = ∅
16	15	iftruei 3568	. . . . . . . . 9 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) = 1o
17		1onn 6587	. . . . . . . . 9 ⊢ 1o ∈ ω
18	16, 17	eqeltri 2269	. . . . . . . 8 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω
19	18	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω)
20	7, 12, 14, 19	fvmptd 5645	. . . . . 6 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
21	20, 16	eqtrdi 2245	. . . . 5 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = 1o)
22	21	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = 1o)
23		fveq1 5560	. . . . . 6 ⊢ ((𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅) → ((𝑆‘𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
24	23	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
25	15	a1i 9	. . . . . . 7 ⊢ (𝑥 = ∅ → ∅ = ∅)
26		eqid 2196	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
27	25, 26	fvmptg 5640	. . . . . 6 ⊢ ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
28	13, 13, 27	mp2an 426	. . . . 5 ⊢ ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
29	24, 28	eqtrdi 2245	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = ∅)
30	22, 29	eqtr3d 2231	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31		1n0 6499	. . . . 5 ⊢ 1o ≠ ∅
32	31	neii 2369	. . . 4 ⊢ ¬ 1o = ∅
33	32	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
34	30, 33	pm2.65da 662	. 2 ⊢ (𝐴 ∈ ℕ∞ → ¬ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅))
35	34	neqned 2374	1 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∅c0 3451  ifcif 3562  ∪ cuni 3840   ↦ cmpt 4095  ωcom 4627  ‘cfv 5259  1_oc1o 6476  ℕ_∞xnninf 7194

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1o 6483

This theorem is referenced by:  exmidsbthrlem  15753