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Theorem aovov0bi 28027
 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 (()) (())

Proof of Theorem aovov0bi
StepHypRef Expression
1 df-ov 6076 . . 3
21eqeq1i 2442 . 2
3 afvfv0bi 27983 . 2 ''' '''
4 df-aov 27943 . . . . 5 (()) '''
54eqeq1i 2442 . . . 4 (()) '''
65bicomi 194 . . 3 ''' (())
74eqeq1i 2442 . . . 4 (()) '''
87bicomi 194 . . 3 ''' (())
96, 8orbi12i 508 . 2 ''' ''' (()) (())
102, 3, 93bitri 263 1 (()) (())
 Colors of variables: wff set class Syntax hints:   wb 177   wo 358   wceq 1652  cvv 2948  c0 3620  cop 3809  cfv 5446  (class class class)co 6073  '''cafv 27939   ((caov 27940 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-dfat 27941  df-afv 27942  df-aov 27943
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