Theorem intsng 3962

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 3748	. . 3 ⊢ {A} = {A, A}
2	1	inteqi 3931	. 2 ⊢ ∩{A} = ∩{A, A}
3		intprg 3961	. . . 4 ⊢ ((A ∈ V ∧ A ∈ V) → ∩{A, A} = (A ∩ A))
4	3	anidms 626	. . 3 ⊢ (A ∈ V → ∩{A, A} = (A ∩ A))
5		inidm 3465	. . 3 ⊢ (A ∩ A) = A
6	4, 5	syl6eq 2401	. 2 ⊢ (A ∈ V → ∩{A, A} = A)
7	2, 6	syl5eq 2397	1 ⊢ (A ∈ V → ∩{A} = A)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ∩ cin 3209  {csn 3738  {cpr 3739  ∩cint 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-sn 3742  df-pr 3743  df-int 3928

This theorem is referenced by:  intsn  3963