Theorem bnj923 35399

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 4067	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2878	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ∖ cdif 4045  ∅c0 4278  {csn 4583  ωcom 7877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047

This theorem is used by:  bnj1098  35414  bnj544  35524  bnj546  35526  bnj594  35542  bnj580  35543  bnj966  35574  bnj967  35575  bnj970  35577  bnj1001  35589  bnj1053  35606  bnj1071  35607  bnj1118  35614  bnj1128  35620  bnj1145  35623