Theorem bnj158 34912

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 2829	. . 3 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (ω ∖ {∅}))
3		eldifsn 4744	. . 3 ⊢ (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
4	2, 3	bitri 275	. 2 ⊢ (𝑚 ∈ 𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5		nnsuc 7838	. 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 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ∖ cdif 3900  ∅c0 4287  {csn 4582  suc csuc 6329  ωcom 7820

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-om 7821

This theorem is referenced by:  bnj168  34913  bnj600  35101  bnj986  35137