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Theorem aov0ov0 27206
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 27162 . 2  |-  ( ( F''' <. A ,  B >. )  =  (/)  ->  ( F `  <. A ,  B >. )  =  (/) )
2 df-aov 27124 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eqeq1i 2323 . 2  |-  ( (( A F B))  =  (/)  <->  ( F''' <. A ,  B >. )  =  (/) )
4 df-ov 5903 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eqeq1i 2323 . 2  |-  ( ( A F B )  =  (/)  <->  ( F `  <. A ,  B >. )  =  (/) )
61, 3, 53imtr4i 257 1  |-  ( (( A F B))  =  (/)  ->  ( A F B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633   (/)c0 3489   <.cop 3677   ` cfv 5292  (class class class)co 5900  '''cafv 27120   ((caov 27121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-nul 3490  df-if 3600  df-fv 5300  df-ov 5903  df-afv 27123  df-aov 27124
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