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Theorem iso0 5817
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 5498 . 2  |-  (/) : (/) -1-1-onto-> (/)
2 ral0 3524 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <-> 
( (/) `  x ) S ( (/) `  y
) )
3 df-isom 5225 . 2  |-  ( (/)  Isom 
R ,  S  (
(/) ,  (/) )  <->  ( (/) : (/) -1-1-onto-> (/)  /\  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <->  ( (/) `  x
) S ( (/) `  y ) ) ) )
41, 2, 3mpbir2an 942 1  |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   A.wral 2455   (/)c0 3422   class class class wbr 4003   -1-1-onto->wf1o 5215   ` cfv 5216    Isom wiso 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-isom 5225
This theorem is referenced by:  zfz1iso  10816
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