Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  elimasn Structured version   GIF version

Theorem elimasn 4619
 Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
elimasn.1 B V
elimasn.2 𝐶 V
Assertion
Ref Expression
elimasn (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A)

Proof of Theorem elimasn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elimasn.2 . . 3 𝐶 V
2 breq2 3742 . . 3 (x = 𝐶 → (BAxBA𝐶))
3 elimasn.1 . . . 4 B V
4 imasng 4617 . . . 4 (B V → (A “ {B}) = {xBAx})
53, 4ax-mp 7 . . 3 (A “ {B}) = {xBAx}
61, 2, 5elab2 2667 . 2 (𝐶 (A “ {B}) ↔ BA𝐶)
7 df-br 3739 . 2 (BA𝐶 ↔ ⟨B, 𝐶 A)
86, 7bitri 173 1 (𝐶 (A “ {B}) ↔ ⟨B, 𝐶 A)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  Vcvv 2535  {csn 3350  ⟨cop 3353   class class class wbr 3738   “ cima 4275 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285 This theorem is referenced by:  elimasng  4620  dfco2  4747  dfco2a  4748  ressn  4785
 Copyright terms: Public domain W3C validator