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Theorem peano3nninf 15110
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 5526 . . . . . . . . . 10 (𝑝 = 𝐴 → (𝑝 𝑖) = (𝐴 𝑖))
21ifeq2d 3564 . . . . . . . . 9 (𝑝 = 𝐴 → if(𝑖 = ∅, 1o, (𝑝 𝑖)) = if(𝑖 = ∅, 1o, (𝐴 𝑖)))
32mpteq2dv 4106 . . . . . . . 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 4604 . . . . . . . . 9 ω ∈ V
65mptex 5755 . . . . . . . 8 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴 𝑖))) ∈ V
73, 4, 6fvmpt 5606 . . . . . . 7 (𝐴 ∈ ℕ → (𝑆𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, (𝐴 𝑖))))
8 eqeq1 2194 . . . . . . . . 9 (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9 unieq 3830 . . . . . . . . . 10 (𝑖 = ∅ → 𝑖 = ∅)
109fveq2d 5531 . . . . . . . . 9 (𝑖 = ∅ → (𝐴 𝑖) = (𝐴 ∅))
118, 10ifbieq2d 3570 . . . . . . . 8 (𝑖 = ∅ → if(𝑖 = ∅, 1o, (𝐴 𝑖)) = if(∅ = ∅, 1o, (𝐴 ∅)))
1211adantl 277 . . . . . . 7 ((𝐴 ∈ ℕ𝑖 = ∅) → if(𝑖 = ∅, 1o, (𝐴 𝑖)) = if(∅ = ∅, 1o, (𝐴 ∅)))
13 peano1 4605 . . . . . . . 8 ∅ ∈ ω
1413a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → ∅ ∈ ω)
15 eqid 2187 . . . . . . . . . 10 ∅ = ∅
1615iftruei 3552 . . . . . . . . 9 if(∅ = ∅, 1o, (𝐴 ∅)) = 1o
17 1onn 6535 . . . . . . . . 9 1o ∈ ω
1816, 17eqeltri 2260 . . . . . . . 8 if(∅ = ∅, 1o, (𝐴 ∅)) ∈ ω
1918a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → if(∅ = ∅, 1o, (𝐴 ∅)) ∈ ω)
207, 12, 14, 19fvmptd 5610 . . . . . 6 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = if(∅ = ∅, 1o, (𝐴 ∅)))
2120, 16eqtrdi 2236 . . . . 5 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = 1o)
2221adantr 276 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = 1o)
23 fveq1 5526 . . . . . 6 ((𝑆𝐴) = (𝑥 ∈ ω ↦ ∅) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2423adantl 277 . . . . 5 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2515a1i 9 . . . . . . 7 (𝑥 = ∅ → ∅ = ∅)
26 eqid 2187 . . . . . . 7 (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
2725, 26fvmptg 5605 . . . . . 6 ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
2813, 13, 27mp2an 426 . . . . 5 ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
2924, 28eqtrdi 2236 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ∅)
3022, 29eqtr3d 2222 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1o = ∅)
31 1n0 6447 . . . . 5 1o ≠ ∅
3231neii 2359 . . . 4 ¬ 1o = ∅
3332a1i 9 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1o = ∅)
3430, 33pm2.65da 662 . 2 (𝐴 ∈ ℕ → ¬ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅))
3534neqned 2364 1 (𝐴 ∈ ℕ → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104   = wceq 1363  wcel 2158  wne 2357  c0 3434  ifcif 3546   cuni 3821  cmpt 4076  ωcom 4601  cfv 5228  1oc1o 6424  xnninf 7132
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1o 6431
This theorem is referenced by:  exmidsbthrlem  15124
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