Users' Mathboxes Mathbox for Steve Rodriguez < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iso0 Structured version   Unicode version

Theorem iso0 27625
Description: The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
Assertion
Ref Expression
iso0  |-  (/)  Isom  R ,  S  ( (/) ,  (/) )

Proof of Theorem iso0
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1o0 5747 . 2  |-  (/) : (/) -1-1-onto-> (/)
2 ral0 3760 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <-> 
( (/) `  x ) S ( (/) `  y
) )
3 df-isom 5498 . 2  |-  ( (/)  Isom 
R ,  S  (
(/) ,  (/) )  <->  ( (/) : (/) -1-1-onto-> (/)  /\  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <->  ( (/) `  x
) S ( (/) `  y ) ) ) )
41, 2, 3mpbir2an 888 1  |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   A.wral 2712   (/)c0 3616   class class class wbr 4243   -1-1-onto->wf1o 5488   ` cfv 5489    Isom wiso 5490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-isom 5498
  Copyright terms: Public domain W3C validator