Theorem intsng 4913

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 4582	. . 3 ⊢ {𝐴} = {𝐴, 𝐴}
2	1	inteqi 4882	. 2 ⊢ ∩ {𝐴} = ∩ {𝐴, 𝐴}
3		intprg 4912	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
4	3	anidms 569	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = (𝐴 ∩ 𝐴))
5		inidm 4197	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
6	4, 5	syl6eq 2874	. 2 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴, 𝐴} = 𝐴)
7	2, 6	syl5eq 2870	1 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∩ cin 3937  {csn 4569  {cpr 4571  ∩ cint 4878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-un 3943  df-in 3945  df-sn 4570  df-pr 4572  df-int 4879

This theorem is referenced by:  intsn  4914  riinint  5841  bj-snmoore  34407  bj-prmoore  34409  elrfi  39298