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Theorem bnj158 30051
 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 2680 . . 3 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
3 eldifsn 4260 . . 3 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
42, 3bitri 263 . 2 (𝑚𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5 nnsuc 6974 . 2 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
64, 5sylbi 206 1 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∖ cdif 3537  ∅c0 3874  {csn 4125  suc csuc 5642  ωcom 6957 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-om 6958 This theorem is referenced by:  bnj168  30052  bnj600  30243  bnj986  30278
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