Theorem compl0 4072

Description: The complement of the empty set is the universe. (Contributed by SF, 2-Jan-2018.)

Assertion

Ref	Expression
compl0	⊢ ∼ ∅ = V

Proof of Theorem compl0

Step	Hyp	Ref	Expression
1		complV 4071	. . 3 ⊢ ∼ V = ∅
2	1	compleqi 3245	. 2 ⊢ ∼ ∼ V = ∼ ∅
3		dblcompl 3228	. 2 ⊢ ∼ ∼ V = V
4	2, 3	eqtr3i 2375	1 ⊢ ∼ ∅ = V

Syntax hints:   = wceq 1642  Vcvv 2860   ∼ ccompl 3206  ∅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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)