Theorem rabxm 4338

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 3425	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
2		exmidd 895	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
3	1, 2	mprg 3051	. 2 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4		unrab 4263	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
5	3, 4	eqtr4i 2756	1 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})

Syntax hints:  ¬ wn 3   ∨ wo 847   = wceq 1541   ∈ wcel 2110  {crab 3393   ∪ cun 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-12 2179  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rab 3394  df-v 3436  df-un 3905

This theorem is referenced by:  elnelun  4341  vtxdgoddnumeven  29525  esumrnmpt2  34071  ddemeas  34239  ballotth  34541  mbfposadd  37686  jm2.22  43007