Theorem bnj158 34705

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 2836	. . 3 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (ω ∖ {∅}))
3		eldifsn 4811	. . 3 ⊢ (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
4	2, 3	bitri 275	. 2 ⊢ (𝑚 ∈ 𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5		nnsuc 7921	. 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 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076   ∖ cdif 3973  ∅c0 4352  {csn 4648  suc csuc 6397  ωcom 7903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-om 7904

This theorem is referenced by:  bnj168  34706  bnj600  34895  bnj986  34931