Theorem bnj923 34761

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 4141	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2857	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106   ∖ cdif 3960  ∅c0 4339  {csn 4631  ωcom 7887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966

This theorem is referenced by:  bnj1098  34776  bnj544  34887  bnj546  34889  bnj594  34905  bnj580  34906  bnj966  34937  bnj967  34938  bnj970  34940  bnj1001  34952  bnj1053  34969  bnj1071  34970  bnj1118  34977  bnj1128  34983  bnj1145  34986