Theorem intsng 4711

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 4390	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 4680	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4710	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 558	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 4026	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	syl6eq 2863	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	syl5eq 2859	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1637   ∈ wcel 2157   ∩ cin 3775  {csn 4377  {cpr 4379  ∩ cint 4676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-v 3400  df-un 3781  df-in 3783  df-sn 4378  df-pr 4380  df-int 4677

This theorem is referenced by:  intsn  4712  riinint  5590  bj-snmoore  33381  elrfi  37760