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Theorem aovpcov0 28032
Description: If the alternative value of the operation on an ordered pair is the universal class, the operation's value at this ordered pair is the empty set. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
aovpcov0  |-  ( (( A F B))  =  _V  ->  ( A F B )  =  (/) )

Proof of Theorem aovpcov0
StepHypRef Expression
1 afvpcfv0 27988 . 2  |-  ( ( F''' <. A ,  B >. )  =  _V  ->  ( F `  <. A ,  B >. )  =  (/) )
2 df-aov 27954 . . 3  |- (( A F B))  =  ( F''' <. A ,  B >. )
32eqeq1i 2445 . 2  |-  ( (( A F B))  =  _V  <->  ( F''' <. A ,  B >. )  =  _V )
4 df-ov 6086 . . 3  |-  ( A F B )  =  ( F `  <. A ,  B >. )
54eqeq1i 2445 . 2  |-  ( ( A F B )  =  (/)  <->  ( F `  <. A ,  B >. )  =  (/) )
61, 3, 53imtr4i 259 1  |-  ( (( A F B))  =  _V  ->  ( A F B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2958   (/)c0 3630   <.cop 3819   ` cfv 5456  (class class class)co 6083  '''cafv 27950   ((caov 27951
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 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-dfat 27952  df-afv 27953  df-aov 27954
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