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Theorem peano3nninf 13262
Description: The successor function on 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 5420 . . . . . . . . . 10 (𝑝 = 𝐴 → (𝑝 𝑖) = (𝐴 𝑖))
21ifeq2d 3490 . . . . . . . . 9 (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑖 = ∅, 1o, (𝐴 𝑖)))
32mpteq2dv 4019 . . . . . . . 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 4507 . . . . . . . . 9 ω ∈ V
65mptex 5646 . . . . . . . 8 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴 𝑖))) ∈ V
73, 4, 6fvmpt 5498 . . . . . . 7 (𝐴 ∈ ℕ → (𝑆𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴 𝑖))))
8 eqeq1 2146 . . . . . . . . 9 (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9 unieq 3745 . . . . . . . . . 10 (𝑖 = ∅ → 𝑖 = ∅)
109fveq2d 5425 . . . . . . . . 9 (𝑖 = ∅ → (𝐴 𝑖) = (𝐴 ∅))
118, 10ifbieq2d 3496 . . . . . . . 8 (𝑖 = ∅ → if(𝑖 = ∅, 1o, (𝐴 𝑖)) = if(∅ = ∅, 1o, (𝐴 ∅)))
1211adantl 275 . . . . . . 7 ((𝐴 ∈ ℕ𝑖 = ∅) → if(𝑖 = ∅, 1o, (𝐴 𝑖)) = if(∅ = ∅, 1o, (𝐴 ∅)))
13 peano1 4508 . . . . . . . 8 ∅ ∈ ω
1413a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → ∅ ∈ ω)
15 eqid 2139 . . . . . . . . . 10 ∅ = ∅
1615iftruei 3480 . . . . . . . . 9 if(∅ = ∅, 1o, (𝐴 ∅)) = 1o
17 1onn 6416 . . . . . . . . 9 1o ∈ ω
1816, 17eqeltri 2212 . . . . . . . 8 if(∅ = ∅, 1o, (𝐴 ∅)) ∈ ω
1918a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → if(∅ = ∅, 1o, (𝐴 ∅)) ∈ ω)
207, 12, 14, 19fvmptd 5502 . . . . . 6 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴 ∅)))
2120, 16syl6eq 2188 . . . . 5 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = 1o)
2221adantr 274 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = 1o)
23 fveq1 5420 . . . . . 6 ((𝑆𝐴) = (𝑥 ∈ ω ↦ ∅) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2423adantl 275 . . . . 5 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2515a1i 9 . . . . . . 7 (𝑥 = ∅ → ∅ = ∅)
26 eqid 2139 . . . . . . 7 (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
2725, 26fvmptg 5497 . . . . . 6 ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
2813, 13, 27mp2an 422 . . . . 5 ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
2924, 28syl6eq 2188 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ∅)
3022, 29eqtr3d 2174 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31 1n0 6329 . . . . 5 1o ≠ ∅
3231neii 2310 . . . 4 ¬ 1o = ∅
3332a1i 9 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
3430, 33pm2.65da 650 . 2 (𝐴 ∈ ℕ → ¬ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅))
3534neqned 2315 1 (𝐴 ∈ ℕ → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1331  wcel 1480  wne 2308  c0 3363  ifcif 3474   cuni 3736  cmpt 3989  ωcom 4504  cfv 5123  1oc1o 6306  xnninf 7005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313
This theorem is referenced by:  exmidsbthrlem  13278
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