Theorem intsng 4004

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 3723	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 3974	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4003	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 401	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 3440	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	eqtrdi 2287	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	eqtrid 2283	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209   ∩ cin 3219  {csn 3709  {cpr 3710  ∩ cint 3970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-sn 3715  df-pr 3716  df-int 3971

This theorem is used by:  intsn  4005  op1stbg  4625  riinint  5043