Theorem peano3nninf 16018

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 5582	. . . . . . . . . 10 ⊢ (𝑝 = 𝐴 → (𝑝‘∪ 𝑖) = (𝐴‘∪ 𝑖))
2	1	ifeq2d 3590	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)))
3	2	mpteq2dv 4139	. . . . . . . 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 4645	. . . . . . . . 9 ⊢ ω ∈ V
6	5	mptex 5817	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))) ∈ V
7	3, 4, 6	fvmpt 5663	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))))
8		eqeq1 2213	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9		unieq 3861	. . . . . . . . . 10 ⊢ (𝑖 = ∅ → ∪ 𝑖 = ∪ ∅)
10	9	fveq2d 5587	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝐴‘∪ 𝑖) = (𝐴‘∪ ∅))
11	8, 10	ifbieq2d 3596	. . . . . . . 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 4646	. . . . . . . 8 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → ∅ ∈ ω)
15		eqid 2206	. . . . . . . . . 10 ⊢ ∅ = ∅
16	15	iftruei 3578	. . . . . . . . 9 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) = 1o
17		1onn 6613	. . . . . . . . 9 ⊢ 1o ∈ ω
18	16, 17	eqeltri 2279	. . . . . . . 8 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω
19	18	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω)
20	7, 12, 14, 19	fvmptd 5667	. . . . . 6 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
21	20, 16	eqtrdi 2255	. . . . 5 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = 1o)
22	21	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = 1o)
23		fveq1 5582	. . . . . 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 2206	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
27	25, 26	fvmptg 5662	. . . . . 6 ⊢ ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
28	13, 13, 27	mp2an 426	. . . . 5 ⊢ ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
29	24, 28	eqtrdi 2255	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = ∅)
30	22, 29	eqtr3d 2241	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31		1n0 6525	. . . . 5 ⊢ 1o ≠ ∅
32	31	neii 2379	. . . 4 ⊢ ¬ 1o = ∅
33	32	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
34	30, 33	pm2.65da 663	. 2 ⊢ (𝐴 ∈ ℕ∞ → ¬ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅))
35	34	neqned 2384	1 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∅c0 3461  ifcif 3572  ∪ cuni 3852   ↦ cmpt 4109  ωcom 4642  ‘cfv 5276  1_oc1o 6502  ℕ_∞xnninf 7228

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-1o 6509

This theorem is referenced by:  exmidsbthrlem  16035