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Theorem bnj923 31145
 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 (𝑛𝐷𝑛 ∈ ω)

Proof of Theorem bnj923
StepHypRef Expression
1 eldifi 3875 . 2 (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2 bnj923.1 . 2 𝐷 = (ω ∖ {∅})
31, 2eleq2s 2857 1 (𝑛𝐷𝑛 ∈ ω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139   ∖ cdif 3712  ∅c0 4058  {csn 4321  ωcom 7230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718 This theorem is referenced by:  bnj1098  31161  bnj544  31271  bnj546  31273  bnj594  31289  bnj580  31290  bnj966  31321  bnj967  31322  bnj970  31324  bnj1001  31335  bnj1053  31351  bnj1071  31352  bnj1118  31359  bnj1128  31365  bnj1145  31368
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