 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iinxsng GIF version

Theorem iinxsng 4042
 Description: A singleton index picks out an instance of an indexed intersection's argument. (Contributed by NM, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Hypothesis
Ref Expression
iinxsng.1 (x = AB = C)
Assertion
Ref Expression
iinxsng (A Vx {A}B = C)
Distinct variable groups:   x,A   x,C
Allowed substitution hints:   B(x)   V(x)

Proof of Theorem iinxsng
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-iin 3972 . 2 x {A}B = {y x {A}y B}
2 iinxsng.1 . . . . 5 (x = AB = C)
32eleq2d 2420 . . . 4 (x = A → (y By C))
43ralsng 3765 . . 3 (A V → (x {A}y By C))
54abbi1dv 2469 . 2 (A V → {y x {A}y B} = C)
61, 5syl5eq 2397 1 (A Vx {A}B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2614  {csn 3737  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-sbc 3047  df-sn 3741  df-iin 3972 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator