Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj923 Structured version   Visualization version   GIF version

Theorem bnj923 31355
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 3930 . 2 (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
2 bnj923.1 . 2 𝐷 = (ω ∖ {∅})
31, 2eleq2s 2896 1 (𝑛𝐷𝑛 ∈ ω)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cdif 3766  c0 4115  {csn 4368  ωcom 7299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772
This theorem is referenced by:  bnj1098  31371  bnj544  31481  bnj546  31483  bnj594  31499  bnj580  31500  bnj966  31531  bnj967  31532  bnj970  31534  bnj1001  31545  bnj1053  31561  bnj1071  31562  bnj1118  31569  bnj1128  31575  bnj1145  31578
  Copyright terms: Public domain W3C validator