Theorem bnj923 35166

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 4084	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2880	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142   ∖ cdif 3901  ∅c0 4285  {csn 4588  ωcom 7860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907

This theorem is used by:  bnj1098  35181  bnj544  35291  bnj546  35293  bnj594  35309  bnj580  35310  bnj966  35341  bnj967  35342  bnj970  35344  bnj1001  35356  bnj1053  35373  bnj1071  35374  bnj1118  35381  bnj1128  35387  bnj1145  35390