Theorem bnj923 35466

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj923.1	⊢ 𝐷 = (ω ∖ {∅})

Assertion

Ref	Expression
bnj923	⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

Proof of Theorem bnj923

Step	Hyp	Ref	Expression
1		eldifi 4070	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2882	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ∖ cdif 4048  ∅c0 4281  {csn 4588  ωcom 7898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-dif 4050

This theorem is used by:  bnj1098  35481  bnj544  35591  bnj546  35593  bnj594  35609  bnj580  35610  bnj966  35641  bnj967  35642  bnj970  35644  bnj1001  35656  bnj1053  35673  bnj1071  35674  bnj1118  35681  bnj1128  35687  bnj1145  35690