Theorem bnj158 34864

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 2827	. . 3 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (ω ∖ {∅}))
3		eldifsn 4741	. . 3 ⊢ (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
4	2, 3	bitri 275	. 2 ⊢ (𝑚 ∈ 𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5		nnsuc 7826	. 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 2931  ∃wrex 3059   ∖ cdif 3897  ∅c0 4284  {csn 4579  suc csuc 6318  ωcom 7808

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 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376  ax-un 7680

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 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-om 7809

This theorem is referenced by:  bnj168  34865  bnj600  35054  bnj986  35090