Theorem peano3nninf 16403

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 5628	. . . . . . . . . 10 ⊢ (𝑝 = 𝐴 → (𝑝‘∪ 𝑖) = (𝐴‘∪ 𝑖))
2	1	ifeq2d 3621	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)))
3	2	mpteq2dv 4175	. . . . . . . 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 4685	. . . . . . . . 9 ⊢ ω ∈ V
6	5	mptex 5869	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))) ∈ V
7	3, 4, 6	fvmpt 5713	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))))
8		eqeq1 2236	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9		unieq 3897	. . . . . . . . . 10 ⊢ (𝑖 = ∅ → ∪ 𝑖 = ∪ ∅)
10	9	fveq2d 5633	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝐴‘∪ 𝑖) = (𝐴‘∪ ∅))
11	8, 10	ifbieq2d 3627	. . . . . . . 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 4686	. . . . . . . 8 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → ∅ ∈ ω)
15		eqid 2229	. . . . . . . . . 10 ⊢ ∅ = ∅
16	15	iftruei 3608	. . . . . . . . 9 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) = 1o
17		1onn 6674	. . . . . . . . 9 ⊢ 1o ∈ ω
18	16, 17	eqeltri 2302	. . . . . . . 8 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω
19	18	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω)
20	7, 12, 14, 19	fvmptd 5717	. . . . . 6 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
21	20, 16	eqtrdi 2278	. . . . 5 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = 1o)
22	21	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = 1o)
23		fveq1 5628	. . . . . 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 2229	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
27	25, 26	fvmptg 5712	. . . . . 6 ⊢ ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
28	13, 13, 27	mp2an 426	. . . . 5 ⊢ ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
29	24, 28	eqtrdi 2278	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = ∅)
30	22, 29	eqtr3d 2264	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31		1n0 6586	. . . . 5 ⊢ 1o ≠ ∅
32	31	neii 2402	. . . 4 ⊢ ¬ 1o = ∅
33	32	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
34	30, 33	pm2.65da 665	. 2 ⊢ (𝐴 ∈ ℕ∞ → ¬ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅))
35	34	neqned 2407	1 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∅c0 3491  ifcif 3602  ∪ cuni 3888   ↦ cmpt 4145  ωcom 4682  ‘cfv 5318  1_oc1o 6561  ℕ_∞xnninf 7294

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6568

This theorem is referenced by:  exmidsbthrlem  16420