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

Theorem opelrn 4484
 Description: Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.)
Hypotheses
Ref Expression
brelrn.1 A V
brelrn.2 B V
Assertion
Ref Expression
opelrn (⟨A, B 𝐶B ran 𝐶)

Proof of Theorem opelrn
StepHypRef Expression
1 df-br 3729 . 2 (A𝐶B ↔ ⟨A, B 𝐶)
2 brelrn.1 . . 3 A V
3 brelrn.2 . . 3 B V
42, 3brelrn 4483 . 2 (A𝐶BB ran 𝐶)
51, 4sylbir 125 1 (⟨A, B 𝐶B ran 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1367  Vcvv 2527  ⟨cop 3343   class class class wbr 3728  ran crn 4262 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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-cnv 4269  df-dm 4271  df-rn 4272 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator