Theorem intsng 4947

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 4602	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 4914	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4945	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 566	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 4190	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	eqtrdi 2780	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	eqtrid 2776	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3913  {csn 4589  {cpr 4591  ∩ cint 4910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-in 3921  df-sn 4590  df-pr 4592  df-int 4911

This theorem is referenced by:  intsn  4948  riinint  5935  bj-snmoore  37101  bj-prmoore  37103  elrfi  42682