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Theorem bnj923 29893
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 3688 . 2 (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2 bnj923.1 . 2 𝐷 = (ω ∖ {∅})
31, 2eleq2s 2700 1 (𝑛𝐷𝑛 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  cdif 3531  c0 3868  {csn 4119  ωcom 6929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-dif 3537
This theorem is referenced by:  bnj1098  29909  bnj544  30019  bnj546  30021  bnj594  30037  bnj580  30038  bnj966  30069  bnj967  30070  bnj970  30072  bnj1001  30083  bnj1053  30099  bnj1071  30100  bnj1118  30107  bnj1128  30113  bnj1145  30116
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