Theorem peano3nninf 16924

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 5674	. . . . . . . . . 10 ⊢ (𝑝 = 𝐴 → (𝑝‘∪ 𝑖) = (𝐴‘∪ 𝑖))
2	1	ifeq2d 3645	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)))
3	2	mpteq2dv 4206	. . . . . . . 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 4720	. . . . . . . . 9 ⊢ ω ∈ V
6	5	mptex 5917	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))) ∈ V
7	3, 4, 6	fvmpt 5759	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))))
8		eqeq1 2241	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9		unieq 3928	. . . . . . . . . 10 ⊢ (𝑖 = ∅ → ∪ 𝑖 = ∪ ∅)
10	9	fveq2d 5679	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝐴‘∪ 𝑖) = (𝐴‘∪ ∅))
11	8, 10	ifbieq2d 3651	. . . . . . . 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 4721	. . . . . . . 8 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → ∅ ∈ ω)
15		eqid 2234	. . . . . . . . . 10 ⊢ ∅ = ∅
16	15	iftruei 3632	. . . . . . . . 9 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) = 1o
17		1onn 6766	. . . . . . . . 9 ⊢ 1o ∈ ω
18	16, 17	eqeltri 2307	. . . . . . . 8 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω
19	18	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω)
20	7, 12, 14, 19	fvmptd 5763	. . . . . 6 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
21	20, 16	eqtrdi 2283	. . . . 5 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = 1o)
22	21	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = 1o)
23		fveq1 5674	. . . . . 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 2234	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
27	25, 26	fvmptg 5758	. . . . . 6 ⊢ ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
28	13, 13, 27	mp2an 426	. . . . 5 ⊢ ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
29	24, 28	eqtrdi 2283	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = ∅)
30	22, 29	eqtr3d 2269	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31		1n0 6678	. . . . 5 ⊢ 1o ≠ ∅
32	31	neii 2416	. . . 4 ⊢ ¬ 1o = ∅
33	32	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
34	30, 33	pm2.65da 667	. 2 ⊢ (𝐴 ∈ ℕ∞ → ¬ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅))
35	34	neqned 2421	1 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   ≠ wne 2414  ∅c0 3512  ifcif 3624  ∪ cuni 3919   ↦ cmpt 4176  ωcom 4717  ‘cfv 5357  1_oc1o 6653  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-coll 4230  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-3an 1007  df-tru 1401  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-reu 2529  df-rab 2531  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-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1o 6660

This theorem is referenced by:  exmidsbthrlem  16941