MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabxm Structured version   Visualization version   GIF version

Theorem rabxm 4390
Description: Law of excluded middle, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
Assertion
Ref Expression
rabxm 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabxm
StepHypRef Expression
1 rabid2im 3469 . . 3 (∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑) → 𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
2 exmidd 896 . . 3 (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑))
31, 2mprg 3067 . 2 𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4 unrab 4315 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
53, 4eqtr4i 2768 1 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 848   = wceq 1540  wcel 2108  {crab 3436  cun 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-un 3956
This theorem is referenced by:  elnelun  4393  vtxdgoddnumeven  29571  esumrnmpt2  34069  ddemeas  34237  ballotth  34540  mbfposadd  37674  jm2.22  43007
  Copyright terms: Public domain W3C validator