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Theorem bnj923 30812
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 3724 . 2 (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2 bnj923.1 . 2 𝐷 = (ω ∖ {∅})
31, 2eleq2s 2717 1 (𝑛𝐷𝑛 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cdif 3564  c0 3907  {csn 4168  ωcom 7050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570
This theorem is referenced by:  bnj1098  30828  bnj544  30938  bnj546  30940  bnj594  30956  bnj580  30957  bnj966  30988  bnj967  30989  bnj970  30991  bnj1001  31002  bnj1053  31018  bnj1071  31019  bnj1118  31026  bnj1128  31032  bnj1145  31035
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