Theorem rabeq0 3573

Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)

Assertion

Ref	Expression
rabeq0	⊢ ({x ∈ A ∣ φ} = ∅ ↔ ∀x ∈ A ¬ φ)

Proof of Theorem rabeq0

Step	Hyp	Ref	Expression
1		ralnex 2625	. 2 ⊢ (∀x ∈ A ¬ φ ↔ ¬ ∃x ∈ A φ)
2		rabn0 3571	. . 3 ⊢ ({x ∈ A ∣ φ} ≠ ∅ ↔ ∃x ∈ A φ)
3	2	necon1bbii 2569	. 2 ⊢ (¬ ∃x ∈ A φ ↔ {x ∈ A ∣ φ} = ∅)
4	1, 3	bitr2i 241	1 ⊢ ({x ∈ A ∣ φ} = ∅ ↔ ∀x ∈ A ¬ φ)

Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642  ∀wral 2615  ∃wrex 2616  {crab 2619  ∅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:  rabnc  3575  nmembers1lem2  6270