Theorem rabxm 4343

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 3432	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∨ ¬ 𝜑) → 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)})
2		exmidd 896	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ∨ ¬ 𝜑))
3	1, 2	mprg 3058	. 2 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
4		unrab 4268	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) = {𝑥 ∈ 𝐴 ∣ (𝜑 ∨ ¬ 𝜑)}
5	3, 4	eqtr4i 2763	1 ⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 𝜑} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})

Syntax hints:  ¬ wn 3   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {crab 3400   ∪ cun 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3401  df-v 3443  df-un 3907

This theorem is referenced by:  elnelun  4346  vtxdgoddnumeven  29610  esumrnmpt2  34206  ddemeas  34374  ballotth  34676  mbfposadd  37839  jm2.22  43273