Theorem intsng 4933

Description: Intersection of a singleton. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
intsng	⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

Proof of Theorem intsng

Step	Hyp	Ref	Expression
1		dfsn2 4586	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 4898	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4929	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 567	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 4165	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	eqtrdi 2792	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	eqtrid 2788	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ∩ cin 3897  {csn 4573  {cpr 4575  ∩ cint 4894

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-v 3443  df-un 3903  df-in 3905  df-sn 4574  df-pr 4576  df-int 4895

This theorem is referenced by:  intsn  4934  riinint  5909  bj-snmoore  35397  bj-prmoore  35399  elrfi  40786