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 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabxm

Step	Hyp	Ref	Expression
1		rabid2im 3469	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
2		exmidd 896	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
3	1, 2	mprg 3067	. 2 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4		unrab 4315	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
5	3, 4	eqtr4i 2768	1 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})

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