Theorem ninexg 4098

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

Assertion

Ref	Expression
ninexg	⊢ ((A ∈ V ∧ B ∈ W) → (A ⩃ B) ∈ V)

Proof of Theorem ninexg

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nineq1 3235	. . 3 ⊢ (x = A → (x ⩃ y) = (A ⩃ y))
2	1	eleq1d 2419	. 2 ⊢ (x = A → ((x ⩃ y) ∈ V ↔ (A ⩃ y) ∈ V))
3		nineq2 3236	. . 3 ⊢ (y = B → (A ⩃ y) = (A ⩃ B))
4	3	eleq1d 2419	. 2 ⊢ (y = B → ((A ⩃ y) ∈ V ↔ (A ⩃ B) ∈ V))
5		ax-nin 4079	. . 3 ⊢ ∃z∀w(w ∈ z ↔ (w ∈ x ⊼ w ∈ y))
6		isset 2864	. . . 4 ⊢ ((x ⩃ y) ∈ V ↔ ∃z z = (x ⩃ y))
7		dfcleq 2347	. . . . . 6 ⊢ (z = (x ⩃ y) ↔ ∀w(w ∈ z ↔ w ∈ (x ⩃ y)))
8		vex 2863	. . . . . . . . 9 ⊢ w ∈ V
9	8	elnin 3225	. . . . . . . 8 ⊢ (w ∈ (x ⩃ y) ↔ (w ∈ x ⊼ w ∈ y))
10	9	bibi2i 304	. . . . . . 7 ⊢ ((w ∈ z ↔ w ∈ (x ⩃ y)) ↔ (w ∈ z ↔ (w ∈ x ⊼ w ∈ y)))
11	10	albii 1566	. . . . . 6 ⊢ (∀w(w ∈ z ↔ w ∈ (x ⩃ y)) ↔ ∀w(w ∈ z ↔ (w ∈ x ⊼ w ∈ y)))
12	7, 11	bitri 240	. . . . 5 ⊢ (z = (x ⩃ y) ↔ ∀w(w ∈ z ↔ (w ∈ x ⊼ w ∈ y)))
13	12	exbii 1582	. . . 4 ⊢ (∃z z = (x ⩃ y) ↔ ∃z∀w(w ∈ z ↔ (w ∈ x ⊼ w ∈ y)))
14	6, 13	bitri 240	. . 3 ⊢ ((x ⩃ y) ∈ V ↔ ∃z∀w(w ∈ z ↔ (w ∈ x ⊼ w ∈ y)))
15	5, 14	mpbir 200	. 2 ⊢ (x ⩃ y) ∈ V
16	2, 4, 15	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → (A ⩃ B) ∈ V)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ⊼ wnan 1287  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⩃ cnin 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  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

This theorem is referenced by:  ninex  4099  complexg  4100  inexg  4101  unexg  4102