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Theorem peano3nninf 11541
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 5288 . . . . . . . . . 10 (𝑝 = 𝐴 → (𝑝 𝑖) = (𝐴 𝑖))
21ifeq2d 3405 . . . . . . . . 9 (𝑝 = 𝐴 → if(𝑖 = ∅, 1𝑜, (𝑝 𝑖)) = if(𝑖 = ∅, 1𝑜, (𝐴 𝑖)))
32mpteq2dv 3921 . . . . . . . 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 4398 . . . . . . . . 9 ω ∈ V
65mptex 5505 . . . . . . . 8 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝐴 𝑖))) ∈ V
73, 4, 6fvmpt 5365 . . . . . . 7 (𝐴 ∈ ℕ → (𝑆𝐴) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1𝑜, (𝐴 𝑖))))
8 eqeq1 2094 . . . . . . . . 9 (𝑖 = ∅ → (𝑖 = ∅ ↔ ∅ = ∅))
9 unieq 3657 . . . . . . . . . 10 (𝑖 = ∅ → 𝑖 = ∅)
109fveq2d 5293 . . . . . . . . 9 (𝑖 = ∅ → (𝐴 𝑖) = (𝐴 ∅))
118, 10ifbieq2d 3411 . . . . . . . 8 (𝑖 = ∅ → if(𝑖 = ∅, 1𝑜, (𝐴 𝑖)) = if(∅ = ∅, 1𝑜, (𝐴 ∅)))
1211adantl 271 . . . . . . 7 ((𝐴 ∈ ℕ𝑖 = ∅) → if(𝑖 = ∅, 1𝑜, (𝐴 𝑖)) = if(∅ = ∅, 1𝑜, (𝐴 ∅)))
13 peano1 4399 . . . . . . . 8 ∅ ∈ ω
1413a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → ∅ ∈ ω)
15 eqid 2088 . . . . . . . . . 10 ∅ = ∅
1615iftruei 3395 . . . . . . . . 9 if(∅ = ∅, 1𝑜, (𝐴 ∅)) = 1𝑜
17 1onn 6259 . . . . . . . . 9 1𝑜 ∈ ω
1816, 17eqeltri 2160 . . . . . . . 8 if(∅ = ∅, 1𝑜, (𝐴 ∅)) ∈ ω
1918a1i 9 . . . . . . 7 (𝐴 ∈ ℕ → if(∅ = ∅, 1𝑜, (𝐴 ∅)) ∈ ω)
207, 12, 14, 19fvmptd 5369 . . . . . 6 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = if(∅ = ∅, 1𝑜, (𝐴 ∅)))
2120, 16syl6eq 2136 . . . . 5 (𝐴 ∈ ℕ → ((𝑆𝐴)‘∅) = 1𝑜)
2221adantr 270 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = 1𝑜)
23 fveq1 5288 . . . . . 6 ((𝑆𝐴) = (𝑥 ∈ ω ↦ ∅) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2423adantl 271 . . . . 5 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ((𝑥 ∈ ω ↦ ∅)‘∅))
2515a1i 9 . . . . . . 7 (𝑥 = ∅ → ∅ = ∅)
26 eqid 2088 . . . . . . 7 (𝑥 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅)
2725, 26fvmptg 5364 . . . . . 6 ((∅ ∈ ω ∧ ∅ ∈ ω) → ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅)
2813, 13, 27mp2an 417 . . . . 5 ((𝑥 ∈ ω ↦ ∅)‘∅) = ∅
2924, 28syl6eq 2136 . . . 4 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ((𝑆𝐴)‘∅) = ∅)
3022, 29eqtr3d 2122 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → 1𝑜 = ∅)
31 1n0 6179 . . . . 5 1𝑜 ≠ ∅
3231neii 2257 . . . 4 ¬ 1𝑜 = ∅
3332a1i 9 . . 3 ((𝐴 ∈ ℕ ∧ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅)) → ¬ 1𝑜 = ∅)
3430, 33pm2.65da 622 . 2 (𝐴 ∈ ℕ → ¬ (𝑆𝐴) = (𝑥 ∈ ω ↦ ∅))
3534neqned 2262 1 (𝐴 ∈ ℕ → (𝑆𝐴) ≠ (𝑥 ∈ ω ↦ ∅))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102   = wceq 1289  wcel 1438  wne 2255  c0 3284  ifcif 3389   cuni 3648  cmpt 3891  ωcom 4395  cfv 5002  1𝑜c1o 6156  xnninf 6768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-1o 6163
This theorem is referenced by:  exmidsbthrlem  11556
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