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Theorem peano3nninf 12785
 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 → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
Distinct variable groups:   𝐴,𝑖,𝑝   𝑆,𝑖,𝑥   𝑥,𝑝
Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑝)

Proof of Theorem peano3nninf
StepHypRef Expression
1 fveq1 5352 . . . . . . . . . 10 (𝑝 = 𝐴 → (𝑝 𝑖) = (𝐴 𝑖))
21ifeq2d 3437 . . . . . . . . 9 (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑖 = ∅, 1o, (𝐴 𝑖)))
32mpteq2dv 3959 . . . . . . . 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 4445 . . . . . . . . 9 ω ∈ V
65mptex 5578 . . . . . . . 8 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴 𝑖))) ∈ V
73, 4, 6fvmpt 5430 . . . . . . 7 (𝐴 ∈ ℕ → (𝑆𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴 𝑖))))
8 eqeq1 2106 . . . . . . . . 9 (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9 unieq 3692 . . . . . . . . . 10 (𝑖 = ∅ → 𝑖 = ∅)
109fveq2d 5357 . . . . . . . . 9 (𝑖 = ∅ → (𝐴 𝑖) = (𝐴 ∅))
118, 10ifbieq2d 3443 . . . . . . . 8 (𝑖 = ∅ → if(𝑖 = ∅, 1o, (𝐴 𝑖)) = if(∅ = ∅, 1o, (𝐴 ∅)))
1211adantl 273 . . . . . . 7 ((𝐴 ∈ ℕ𝑖 = ∅) → if(𝑖 = ∅, 1o, (𝐴 𝑖)) = if(∅ = ∅, 1o, (𝐴 ∅)))
13 peano1 4446 . . . . . . . 8 ∅ ∈ ω
1413a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → ∅ ∈ ω)
15 eqid 2100 . . . . . . . . . 10 ∅ = ∅
1615iftruei 3427 . . . . . . . . 9 if(∅ = ∅, 1o, (𝐴 ∅)) = 1o
17 1onn 6346 . . . . . . . . 9 1o ∈ ω
1816, 17eqeltri 2172 . . . . . . . 8 if(∅ = ∅, 1o, (𝐴 ∅)) ∈ ω
1918a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → if(∅ = ∅, 1o, (𝐴 ∅)) ∈ ω)
207, 12, 14, 19fvmptd 5434 . . . . . 6 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴 ∅)))
2120, 16syl6eq 2148 . . . . 5 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = 1o)
2221adantr 272 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = 1o)
23 fveq1 5352 . . . . . 6 ((𝑆𝐴) = (𝑥 ∈ ω ↦ ∅) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2423adantl 273 . . . . 5 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2515a1i 9 . . . . . . 7 (𝑥 = ∅ → ∅ = ∅)
26 eqid 2100 . . . . . . 7 (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
2725, 26fvmptg 5429 . . . . . 6 ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
2813, 13, 27mp2an 420 . . . . 5 ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
2924, 28syl6eq 2148 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ∅)
3022, 29eqtr3d 2134 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31 1n0 6259 . . . . 5 1o ≠ ∅
3231neii 2269 . . . 4 ¬ 1o = ∅
3332a1i 9 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
3430, 33pm2.65da 628 . 2 (𝐴 ∈ ℕ → ¬ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅))
3534neqned 2274 1 (𝐴 ∈ ℕ → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   = wceq 1299   ∈ wcel 1448   ≠ wne 2267  ∅c0 3310  ifcif 3421  ∪ cuni 3683   ↦ cmpt 3929  ωcom 4442  ‘cfv 5059  1oc1o 6236  ℕ∞xnninf 6917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-iinf 4440 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-1o 6243 This theorem is referenced by:  exmidsbthrlem  12801
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