Theorem bnj923 34783

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 4130	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2858	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ∖ cdif 3947  ∅c0 4332  {csn 4625  ωcom 7888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953

This theorem is referenced by:  bnj1098  34798  bnj544  34909  bnj546  34911  bnj594  34927  bnj580  34928  bnj966  34959  bnj967  34960  bnj970  34962  bnj1001  34974  bnj1053  34991  bnj1071  34992  bnj1118  34999  bnj1128  35005  bnj1145  35008