Theorem bnj923 32039

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 4103	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2931	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∖ cdif 3933  ∅c0 4291  {csn 4567  ωcom 7580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939

This theorem is referenced by:  bnj1098  32055  bnj544  32166  bnj546  32168  bnj594  32184  bnj580  32185  bnj966  32216  bnj967  32217  bnj970  32219  bnj1001  32231  bnj1053  32248  bnj1071  32249  bnj1118  32256  bnj1128  32262  bnj1145  32265