Theorem complV 4071

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

Assertion

Ref	Expression
complV	⊢ ∼ V = ∅

Proof of Theorem complV

Step	Hyp	Ref	Expression
1		compldif 4070	. 2 ⊢ ∼ V = (V ∖ V)
2		df-nul 3552	. 2 ⊢ ∅ = (V ∖ V)
3	1, 2	eqtr4i 2376	1 ⊢ ∼ V = ∅

Syntax hints:   = wceq 1642  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207  ∅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:  compl0  4072  incompl  4074  0ex  4111  nulnnn  4557