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Theorem iso0 36649
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 5847 . 2  |-  (/) : (/) -1-1-onto-> (/)
2 ral0 3873 . 2  |-  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <-> 
( (/) `  x ) S ( (/) `  y
) )
3 df-isom 5590 . 2  |-  ( (/)  Isom 
R ,  S  (
(/) ,  (/) )  <->  ( (/) : (/) -1-1-onto-> (/)  /\  A. x  e.  (/)  A. y  e.  (/)  ( x R y  <->  ( (/) `  x
) S ( (/) `  y ) ) ) )
41, 2, 3mpbir2an 930 1  |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   A.wral 2736   (/)c0 3730   class class class wbr 4401   -1-1-onto->wf1o 5580   ` cfv 5581    Isom wiso 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-isom 5590
This theorem is referenced by: (None)
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