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Theorem bnj158 34743
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 2833 . . 3 (𝑚𝐷𝑚 ∈ (ω ∖ {∅}))
3 eldifsn 4786 . . 3 (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
42, 3bitri 275 . 2 (𝑚𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5 nnsuc 7905 . 2 ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
64, 5sylbi 217 1 (𝑚𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cdif 3948  c0 4333  {csn 4626  suc csuc 6386  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-om 7888
This theorem is referenced by:  bnj168  34744  bnj600  34933  bnj986  34969
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