Theorem bnj923 34744

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 4154	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2862	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  ∅c0 4352  {csn 4648  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

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979

This theorem is referenced by:  bnj1098  34759  bnj544  34870  bnj546  34872  bnj594  34888  bnj580  34889  bnj966  34920  bnj967  34921  bnj970  34923  bnj1001  34935  bnj1053  34952  bnj1071  34953  bnj1118  34960  bnj1128  34966  bnj1145  34969