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Theorem bnj158 30923
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 2722 . . 3 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
3 eldifsn 4350 . . 3 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
42, 3bitri 264 . 2 (𝑚𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5 nnsuc 7124 . 2 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
64, 5sylbi 207 1 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cdif 3604  c0 3948  {csn 4210  suc csuc 5763  ωcom 7107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-om 7108
This theorem is referenced by:  bnj168  30924  bnj600  31115  bnj986  31150
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