Theorem bnj923 35265

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 4081	. 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 1570   ∈ wcel 2145   ∖ cdif 3899  ∅c0 4282  {csn 4587  ωcom 7865

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905

This theorem is used by:  bnj1098  35280  bnj544  35390  bnj546  35392  bnj594  35408  bnj580  35409  bnj966  35440  bnj967  35441  bnj970  35443  bnj1001  35455  bnj1053  35472  bnj1071  35473  bnj1118  35480  bnj1128  35486  bnj1145  35489