Theorem rabxm 3574

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		rabid2 2789	. . 3 ⊢ (A = {x ∈ A ∣ (φ ∨ ¬ φ)} ↔ ∀x ∈ A (φ ∨ ¬ φ))
2		exmid 404	. . . 4 ⊢ (φ ∨ ¬ φ)
3	2	a1i 10	. . 3 ⊢ (x ∈ A → (φ ∨ ¬ φ))
4	1, 3	mprgbir 2685	. 2 ⊢ A = {x ∈ A ∣ (φ ∨ ¬ φ)}
5		unrab 3527	. 2 ⊢ ({x ∈ A ∣ φ} ∪ {x ∈ A ∣ ¬ φ}) = {x ∈ A ∣ (φ ∨ ¬ φ)}
6	4, 5	eqtr4i 2376	1 ⊢ A = ({x ∈ A ∣ φ} ∪ {x ∈ A ∣ ¬ φ})

Syntax hints:  ¬ wn 3   ∨ wo 357   = wceq 1642   ∈ wcel 1710  {crab 2619   ∪ cun 3208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215

This theorem is referenced by: (None)