Theorem rabnc 3575

Description: Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)

Assertion

Ref	Expression
rabnc	⊢ ({x ∈ A ∣ φ} ∩ {x ∈ A ∣ ¬ φ}) = ∅

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem rabnc

Step	Hyp	Ref	Expression
1		inrab 3528	. 2 ⊢ ({x ∈ A ∣ φ} ∩ {x ∈ A ∣ ¬ φ}) = {x ∈ A ∣ (φ ∧ ¬ φ)}
2		rabeq0 3573	. . 3 ⊢ ({x ∈ A ∣ (φ ∧ ¬ φ)} = ∅ ↔ ∀x ∈ A ¬ (φ ∧ ¬ φ))
3		pm3.24 852	. . . 4 ⊢ ¬ (φ ∧ ¬ φ)
4	3	a1i 10	. . 3 ⊢ (x ∈ A → ¬ (φ ∧ ¬ φ))
5	2, 4	mprgbir 2685	. 2 ⊢ {x ∈ A ∣ (φ ∧ ¬ φ)} = ∅
6	1, 5	eqtri 2373	1 ⊢ ({x ∈ A ∣ φ} ∩ {x ∈ A ∣ ¬ φ}) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2619   ∩ cin 3209  ∅c0 3551

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-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)