Theorem inexg 4101

Description: The intersection of two sets is a set. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
inexg	⊢ ((A ∈ V ∧ B ∈ W) → (A ∩ B) ∈ V)

Proof of Theorem inexg

Step	Hyp	Ref	Expression
1		df-in 3214	. 2 ⊢ (A ∩ B) = ∼ (A ⩃ B)
2		ninexg 4098	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (A ⩃ B) ∈ V)
3		complexg 4100	. . 3 ⊢ ((A ⩃ B) ∈ V → ∼ (A ⩃ B) ∈ V)
4	2, 3	syl 15	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → ∼ (A ⩃ B) ∈ V)
5	1, 4	syl5eqel 2437	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ∩ B) ∈ V)

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710  Vcvv 2860   ⩃ cnin 3205   ∼ ccompl 3206   ∩ cin 3209

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  ax-nin 4079

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

This theorem is referenced by:  difexg  4103  inex  4106  xpkexg  4289  imakexg  4300  cokexg  4310  peano5  4410  spfininduct  4541  xpexg  5115  resexg  5117  txpexg  5785  fixexg  5789  clos1induct  5881  frds  5936  pmex  6006  nenpw1pwlem1  6085  ovcelem1  6172  fnfreclem1  6318