Theorem intsng 4942

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 4593	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 4910	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4940	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 577	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 4123	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	eqtrdi 2811	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	eqtrid 2807	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ∩ cin 4089  {csn 4584  {cpr 4586  ∩ cint 4906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-in 4091  df-un 4151  df-sn 4585  df-pr 4587  df-int 4907

This theorem is used by:  intsn  4943  riinint  5953  bj-snmoore  38126  bj-prmoore  38128  elrfi  43798