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Theorem aovov0bi 27729
Description: The operation's value on an ordered pair is the empty set if and only if the alternative value of the operation on this ordered pair is either the empty set or the universal class. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovov0bi  |-  ( ( A F B )  =  (/)  <->  ( (( A F B))  =  (/)  \/ (( A F B))  =  _V ) )

Proof of Theorem aovov0bi
StepHypRef Expression
1 df-ov 6023 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
21eqeq1i 2394 . 2  |-  ( ( A F B )  =  (/)  <->  ( F `  <. A ,  B >. )  =  (/) )
3 afvfv0bi 27685 . 2  |-  ( ( F `  <. A ,  B >. )  =  (/)  <->  (
( F''' <. A ,  B >. )  =  (/)  \/  ( F'''
<. A ,  B >. )  =  _V ) )
4 df-aov 27644 . . . . 5  |- (( A F B))  =  ( F''' <. A ,  B >. )
54eqeq1i 2394 . . . 4  |-  ( (( A F B))  =  (/)  <->  ( F''' <. A ,  B >. )  =  (/) )
65bicomi 194 . . 3  |-  ( ( F''' <. A ,  B >. )  =  (/)  <-> (( A F B))  =  (/) )
74eqeq1i 2394 . . . 4  |-  ( (( A F B))  =  _V  <->  ( F''' <. A ,  B >. )  =  _V )
87bicomi 194 . . 3  |-  ( ( F''' <. A ,  B >. )  =  _V  <-> (( A F B))  =  _V )
96, 8orbi12i 508 . 2  |-  ( ( ( F''' <. A ,  B >. )  =  (/)  \/  ( F'''
<. A ,  B >. )  =  _V )  <->  ( (( A F B))  =  (/)  \/ (( A F B))  =  _V ) )
102, 3, 93bitri 263 1  |-  ( ( A F B )  =  (/)  <->  ( (( A F B))  =  (/)  \/ (( A F B))  =  _V ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    = wceq 1649   _Vcvv 2899   (/)c0 3571   <.cop 3760   ` cfv 5394  (class class class)co 6020  '''cafv 27640   ((caov 27641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-dfat 27642  df-afv 27643  df-aov 27644
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