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Theorem aov0ov0 28034
Description: If the alternative value of the operation on an ordered pair is the empty set, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aov0ov0  |-  ( (( A F B))  =  (/)  ->  ( A F B )  =  (/) )

Proof of Theorem aov0ov0
StepHypRef Expression
1 afv0fv0 27990 . 2  |-  ( ( F''' <. A ,  B >. )  =  (/)  ->  ( F `  <. A ,  B >. )  =  (/) )
2 df-aov 27953 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eqeq1i 2444 . 2  |-  ( (( A F B))  =  (/)  <->  ( F''' <. A ,  B >. )  =  (/) )
4 df-ov 6085 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eqeq1i 2444 . 2  |-  ( ( A F B )  =  (/)  <->  ( F `  <. A ,  B >. )  =  (/) )
61, 3, 53imtr4i 259 1  |-  ( (( A F B))  =  (/)  ->  ( A F B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   (/)c0 3629   <.cop 3818   ` cfv 5455  (class class class)co 6082  '''cafv 27949   ((caov 27950
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-nul 3630  df-if 3741  df-fv 5463  df-ov 6085  df-afv 27952  df-aov 27953
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