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Theorem bnj158 32019
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj158.1 𝐷 = (ω ∖ {∅})
Assertion
Ref Expression
bnj158 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
Distinct variable group:   𝑚,𝑝
Allowed substitution hints:   𝐷(𝑚,𝑝)

Proof of Theorem bnj158
StepHypRef Expression
1 bnj158.1 . . . 4 𝐷 = (ω ∖ {∅})
21eleq2i 2907 . . 3 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
3 eldifsn 4702 . . 3 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
42, 3bitri 278 . 2 (𝑚𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5 nnsuc 7582 . 2 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
64, 5sylbi 220 1 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3013  wrex 3133  cdif 3915  c0 4274  {csn 4548  suc csuc 6176  ωcom 7565
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 5186  ax-nul 5193  ax-pr 5313  ax-un 7446
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 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-tr 5156  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-om 7566
This theorem is referenced by:  bnj168  32020  bnj600  32211  bnj986  32247
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