Theorem peano3nninf 14759

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 5515	. . . . . . . . . 10 ⊢ (𝑝 = 𝐴 → (𝑝‘∪ 𝑖) = (𝐴‘∪ 𝑖))
2	1	ifeq2d 3553	. . . . . . . . 9 ⊢ (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝‘∪ 𝑖)) = if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖)))
3	2	mpteq2dv 4095	. . . . . . . 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 4593	. . . . . . . . 9 ⊢ ω ∈ V
6	5	mptex 5743	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))) ∈ V
7	3, 4, 6	fvmpt 5594	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴‘∪ 𝑖))))
8		eqeq1 2184	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9		unieq 3819	. . . . . . . . . 10 ⊢ (𝑖 = ∅ → ∪ 𝑖 = ∪ ∅)
10	9	fveq2d 5520	. . . . . . . . 9 ⊢ (𝑖 = ∅ → (𝐴‘∪ 𝑖) = (𝐴‘∪ ∅))
11	8, 10	ifbieq2d 3559	. . . . . . . 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 4594	. . . . . . . 8 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → ∅ ∈ ω)
15		eqid 2177	. . . . . . . . . 10 ⊢ ∅ = ∅
16	15	iftruei 3541	. . . . . . . . 9 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) = 1o
17		1onn 6521	. . . . . . . . 9 ⊢ 1o ∈ ω
18	16, 17	eqeltri 2250	. . . . . . . 8 ⊢ if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω
19	18	a1i 9	. . . . . . 7 ⊢ (𝐴 ∈ ℕ∞ → if(∅ = ∅, 1o, (𝐴‘∪ ∅)) ∈ ω)
20	7, 12, 14, 19	fvmptd 5598	. . . . . 6 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴‘∪ ∅)))
21	20, 16	eqtrdi 2226	. . . . 5 ⊢ (𝐴 ∈ ℕ∞ → ((𝑆‘𝐴)‘∅) = 1o)
22	21	adantr 276	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = 1o)
23		fveq1 5515	. . . . . 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 2177	. . . . . . 7 ⊢ (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
27	25, 26	fvmptg 5593	. . . . . 6 ⊢ ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
28	13, 13, 27	mp2an 426	. . . . 5 ⊢ ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
29	24, 28	eqtrdi 2226	. . . 4 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆‘𝐴)‘∅) = ∅)
30	22, 29	eqtr3d 2212	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31		1n0 6433	. . . . 5 ⊢ 1o ≠ ∅
32	31	neii 2349	. . . 4 ⊢ ¬ 1o = ∅
33	32	a1i 9	. . 3 ⊢ ((𝐴 ∈ ℕ∞ ∧ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
34	30, 33	pm2.65da 661	. 2 ⊢ (𝐴 ∈ ℕ∞ → ¬ (𝑆‘𝐴) = (𝑥 ∈ ω ↦ ∅))
35	34	neqned 2354	1 ⊢ (𝐴 ∈ ℕ∞ → (𝑆‘𝐴) ≠ (𝑥 ∈ ω ↦ ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∅c0 3423  ifcif 3535  ∪ cuni 3810   ↦ cmpt 4065  ωcom 4590  ‘cfv 5217  1_oc1o 6410  ℕ_∞xnninf 7118

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417

This theorem is referenced by:  exmidsbthrlem  14773