Theorem bnj158 34712

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	⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)

Distinct variable group:   𝑚,𝑝

Allowed substitution hints:   𝐷(𝑚,𝑝)

Proof of Theorem bnj158

Step	Hyp	Ref	Expression
1		bnj158.1	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
2	1	eleq2i 2820	. . 3 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (ω ∖ {∅}))
3		eldifsn 4737	. . 3 ⊢ (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
4	2, 3	bitri 275	. 2 ⊢ (𝑚 ∈ 𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5		nnsuc 7817	. 2 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
6	4, 5	sylbi 217	1 ⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∖ cdif 3900  ∅c0 4284  {csn 4577  suc csuc 6309  ωcom 7799

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-tr 5200  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-om 7800

This theorem is referenced by:  bnj168  34713  bnj600  34902  bnj986  34938