Theorem bnj923 35102

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 4091	. 2 ⊢ (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2		bnj923.1	. 2 ⊢ 𝐷 = (ω ∖ {∅})
3	1, 2	eleq2s 2887	1 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149   ∖ cdif 3908  ∅c0 4292  {csn 4592  ωcom 7862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3463  df-dif 3914

This theorem is referenced by:  bnj1098  35117  bnj544  35227  bnj546  35229  bnj594  35245  bnj580  35246  bnj966  35277  bnj967  35278  bnj970  35280  bnj1001  35292  bnj1053  35309  bnj1071  35310  bnj1118  35317  bnj1128  35323  bnj1145  35326