Theorem elcomplg 3218

Description: Membership in class complement. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
elcomplg	⊢ (A ∈ V → (A ∈ ∼ B ↔ ¬ A ∈ B))

Proof of Theorem elcomplg

Step	Hyp	Ref	Expression
1		df-compl 3212	. . 3 ⊢ ∼ B = (B ⩃ B)
2	1	eleq2i 2417	. 2 ⊢ (A ∈ ∼ B ↔ A ∈ (B ⩃ B))
3		elning 3217	. . 3 ⊢ (A ∈ V → (A ∈ (B ⩃ B) ↔ (A ∈ B ⊼ A ∈ B)))
4		df-nan 1288	. . . 4 ⊢ ((A ∈ B ⊼ A ∈ B) ↔ ¬ (A ∈ B ∧ A ∈ B))
5		anidm 625	. . . 4 ⊢ ((A ∈ B ∧ A ∈ B) ↔ A ∈ B)
6	4, 5	xchbinx 301	. . 3 ⊢ ((A ∈ B ⊼ A ∈ B) ↔ ¬ A ∈ B)
7	3, 6	syl6bb 252	. 2 ⊢ (A ∈ V → (A ∈ (B ⩃ B) ↔ ¬ A ∈ B))
8	2, 7	syl5bb 248	1 ⊢ (A ∈ V → (A ∈ ∼ B ↔ ¬ A ∈ B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287   ∈ wcel 1710   ⩃ cnin 3204   ∼ ccompl 3205

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 2478  df-v 2861  df-nin 3211  df-compl 3212

This theorem is referenced by:  elin  3219  elun  3220  eldif  3221  elcompl  3225  nnadjoinpw  4521  nmembers1lem3  6270