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Theorem nnn1suc 39397
Description: A positive integer that is not 1 is a successor of some other positive integer. (Contributed by Steven Nguyen, 19-Aug-2023.)
Assertion
Ref Expression
nnn1suc ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem nnn1suc
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neeq1 3076 . . . 4 (𝑦 = 1 → (𝑦 ≠ 1 ↔ 1 ≠ 1))
2 eqeq2 2836 . . . . 5 (𝑦 = 1 → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = 1))
32rexbidv 3289 . . . 4 (𝑦 = 1 → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = 1))
41, 3imbi12d 348 . . 3 (𝑦 = 1 → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ (1 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 1)))
5 neeq1 3076 . . . 4 (𝑦 = 𝑧 → (𝑦 ≠ 1 ↔ 𝑧 ≠ 1))
6 eqeq2 2836 . . . . 5 (𝑦 = 𝑧 → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = 𝑧))
76rexbidv 3289 . . . 4 (𝑦 = 𝑧 → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑧))
85, 7imbi12d 348 . . 3 (𝑦 = 𝑧 → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ (𝑧 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑧)))
9 neeq1 3076 . . . 4 (𝑦 = (𝑧 + 1) → (𝑦 ≠ 1 ↔ (𝑧 + 1) ≠ 1))
10 eqeq2 2836 . . . . 5 (𝑦 = (𝑧 + 1) → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = (𝑧 + 1)))
1110rexbidv 3289 . . . 4 (𝑦 = (𝑧 + 1) → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1)))
129, 11imbi12d 348 . . 3 (𝑦 = (𝑧 + 1) → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ ((𝑧 + 1) ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1))))
13 neeq1 3076 . . . 4 (𝑦 = 𝐴 → (𝑦 ≠ 1 ↔ 𝐴 ≠ 1))
14 eqeq2 2836 . . . . 5 (𝑦 = 𝐴 → ((𝑥 + 1) = 𝑦 ↔ (𝑥 + 1) = 𝐴))
1514rexbidv 3289 . . . 4 (𝑦 = 𝐴 → (∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦 ↔ ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴))
1613, 15imbi12d 348 . . 3 (𝑦 = 𝐴 → ((𝑦 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑦) ↔ (𝐴 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)))
17 df-ne 3015 . . . 4 (1 ≠ 1 ↔ ¬ 1 = 1)
18 eqid 2824 . . . . 5 1 = 1
1918pm2.24i 153 . . . 4 (¬ 1 = 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 1)
2017, 19sylbi 220 . . 3 (1 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 1)
21 id 22 . . . . 5 (𝑧 ∈ ℕ → 𝑧 ∈ ℕ)
22 oveq1 7156 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1))
2322eqeq1d 2826 . . . . . 6 (𝑥 = 𝑧 → ((𝑥 + 1) = (𝑧 + 1) ↔ (𝑧 + 1) = (𝑧 + 1)))
2423adantl 485 . . . . 5 ((𝑧 ∈ ℕ ∧ 𝑥 = 𝑧) → ((𝑥 + 1) = (𝑧 + 1) ↔ (𝑧 + 1) = (𝑧 + 1)))
25 eqidd 2825 . . . . 5 (𝑧 ∈ ℕ → (𝑧 + 1) = (𝑧 + 1))
2621, 24, 25rspcedvd 3612 . . . 4 (𝑧 ∈ ℕ → ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1))
27262a1d 26 . . 3 (𝑧 ∈ ℕ → ((𝑧 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝑧) → ((𝑧 + 1) ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = (𝑧 + 1))))
284, 8, 12, 16, 20, 27nnind 11652 . 2 (𝐴 ∈ ℕ → (𝐴 ≠ 1 → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴))
2928imp 410 1 ((𝐴 ∈ ℕ ∧ 𝐴 ≠ 1) → ∃𝑥 ∈ ℕ (𝑥 + 1) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wne 3014  wrex 3134  (class class class)co 7149  1c1 10536   + caddc 10538  cn 11634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-1cn 10593
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-om 7575  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-nn 11635
This theorem is referenced by: (None)
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