Theorem aovov0bi 38568

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

Step	Hyp	Ref	Expression
1		df-ov 6308	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	eqeq1i 2429	. 2 |- ( ( A F B ) = (/) <-> ( F ` <. A , B >. ) = (/) )
3		afvfv0bi 38524	. 2 |- ( ( F ` <. A , B >. ) = (/) <-> ( ( F''' <. A , B >. ) = (/) \/ ( F''' <. A , B >. ) = _V ) )
4		df-aov 38490	. . . . 5 |- (( A F B)) = ( F''' <. A , B >. )
5	4	eqeq1i 2429	. . . 4 |- ( (( A F B)) = (/) <-> ( F''' <. A , B >. ) = (/) )
6	5	bicomi 205	. . 3 |- ( ( F''' <. A , B >. ) = (/) <-> (( A F B)) = (/) )
7	4	eqeq1i 2429	. . . 4 |- ( (( A F B)) = _V <-> ( F''' <. A , B >. ) = _V )
8	7	bicomi 205	. . 3 |- ( ( F''' <. A , B >. ) = _V <-> (( A F B)) = _V )
9	6, 8	orbi12i 523	. 2 |- ( ( ( F''' <. A , B >. ) = (/) \/ ( F''' <. A , B >. ) = _V ) <-> ( (( A F B)) = (/) \/ (( A F B)) = _V ) )
10	2, 3, 9	3bitri 274	1 |- ( ( A F B ) = (/) <-> ( (( A F B)) = (/) \/ (( A F B)) = _V ) )

Syntax hints:   <-> wb 187   \/ wo 369   = wceq 1437   _Vcvv 3080   (/)c0 3761   <.cop 4004   ` cfv 5601  (class class class)co 6305  '''cafv 38486   ((caov 38487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-dfat 38488  df-afv 38489  df-aov 38490

This theorem is referenced by: (None)