Theorem nincom 3227

Description: Anti-intersection is commutative. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
nincom	⊢ (A ⩃ B) = (B ⩃ A)

Proof of Theorem nincom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nancom 1290	. . 3 ⊢ ((x ∈ A ⊼ x ∈ B) ↔ (x ∈ B ⊼ x ∈ A))
2		vex 2863	. . . 4 ⊢ x ∈ V
3	2	elnin 3225	. . 3 ⊢ (x ∈ (A ⩃ B) ↔ (x ∈ A ⊼ x ∈ B))
4	2	elnin 3225	. . 3 ⊢ (x ∈ (B ⩃ A) ↔ (x ∈ B ⊼ x ∈ A))
5	1, 3, 4	3bitr4i 268	. 2 ⊢ (x ∈ (A ⩃ B) ↔ x ∈ (B ⩃ A))
6	5	eqriv 2350	1 ⊢ (A ⩃ B) = (B ⩃ A)

This proof depends on syntax axioms:   ⊼ wnan 1287   = wceq 1642   ∈ wcel 1710   ⩃ cnin 3205

This proof depends on 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 proof 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 used by:  nineq2  3236