Theorem rint0 3967

Description: Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
rint0	⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Proof of Theorem rint0

Step	Hyp	Ref	Expression
1		inteq 3931	. . 3 ⊢ (𝑋 = ∅ → ∩ 𝑋 = ∩ ∅)
2	1	ineq2d 3408	. 2 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = (𝐴 ∩ ∩ ∅))
3		int0 3942	. . . 4 ⊢ ∩ ∅ = V
4	3	ineq2i 3405	. . 3 ⊢ (𝐴 ∩ ∩ ∅) = (𝐴 ∩ V)
5		inv1 3531	. . 3 ⊢ (𝐴 ∩ V) = 𝐴
6	4, 5	eqtri 2252	. 2 ⊢ (𝐴 ∩ ∩ ∅) = 𝐴
7	2, 6	eqtrdi 2280	1 ⊢ (𝑋 = ∅ → (𝐴 ∩ ∩ 𝑋) = 𝐴)

Syntax hints:   → wi 4   = wceq 1397  Vcvv 2802   ∩ cin 3199  ∅c0 3494  ∩ cint 3928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-int 3929

This theorem is referenced by: (None)