Theorem bnj158 33740

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 2826	. . 3 ⊢ (𝑚 ∈ 𝐷 ↔ 𝑚 ∈ (ω ∖ {∅}))
3		eldifsn 4791	. . 3 ⊢ (𝑚 ∈ (ω ∖ {∅}) ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
4	2, 3	bitri 275	. 2 ⊢ (𝑚 ∈ 𝐷 ↔ (𝑚 ∈ ω ∧ 𝑚 ≠ ∅))
5		nnsuc 7873	. 2 ⊢ ((𝑚 ∈ ω ∧ 𝑚 ≠ ∅) → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
6	4, 5	sylbi 216	1 ⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∃wrex 3071   ∖ cdif 3946  ∅c0 4323  {csn 4629  suc csuc 6367  ωcom 7855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-om 7856

This theorem is referenced by:  bnj168  33741  bnj600  33930  bnj986  33966