Theorem intsng 4882

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 4540	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 4849	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4878	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 570	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 4119	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	eqtrdi 2787	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	syl5eq 2783	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112   ∩ cin 3852  {csn 4527  {cpr 4529  ∩ cint 4845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-v 3400  df-un 3858  df-in 3860  df-sn 4528  df-pr 4530  df-int 4846

This theorem is referenced by:  intsn  4883  riinint  5822  bj-snmoore  34968  bj-prmoore  34970  elrfi  40160