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Theorem rabxm 4106
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 rabid2 3267 . . 3 (𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)} ↔ ∀𝑥𝐴 (𝜑 ∨ ¬ 𝜑))
2 exmidd 881 . . 3 (𝑥𝐴 → (𝜑 ∨ ¬ 𝜑))
31, 2mprgbir 3076 . 2 𝐴 = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4 unrab 4046 . 2 ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}) = {𝑥𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
53, 4eqtr4i 2796 1 𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 836   = wceq 1631  wcel 2145  {crab 3065  cun 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-un 3728
This theorem is referenced by:  elnelun  4109  elnelunOLD  4111  vtxdgoddnumeven  26684  esumrnmpt2  30470  ddemeas  30639  ballotth  30939  mbfposadd  33789  jm2.22  38088
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