Theorem aovov0bi 38833

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 6323	. . 3 |- ( A F B ) = ( F ` <. A , B >. )
2	1	eqeq1i 2467	. 2 |- ( ( A F B ) = (/) <-> ( F ` <. A , B >. ) = (/) )
3		afvfv0bi 38789	. 2 |- ( ( F ` <. A , B >. ) = (/) <-> ( ( F''' <. A , B >. ) = (/) \/ ( F''' <. A , B >. ) = _V ) )
4		df-aov 38754	. . . . 5 |- (( A F B)) = ( F''' <. A , B >. )
5	4	eqeq1i 2467	. . . 4 |- ( (( A F B)) = (/) <-> ( F''' <. A , B >. ) = (/) )
6	5	bicomi 207	. . 3 |- ( ( F''' <. A , B >. ) = (/) <-> (( A F B)) = (/) )
7	4	eqeq1i 2467	. . . 4 |- ( (( A F B)) = _V <-> ( F''' <. A , B >. ) = _V )
8	7	bicomi 207	. . 3 |- ( ( F''' <. A , B >. ) = _V <-> (( A F B)) = _V )
9	6, 8	orbi12i 528	. 2 |- ( ( ( F''' <. A , B >. ) = (/) \/ ( F''' <. A , B >. ) = _V ) <-> ( (( A F B)) = (/) \/ (( A F B)) = _V ) )
10	2, 3, 9	3bitri 279	1 |- ( ( A F B ) = (/) <-> ( (( A F B)) = (/) \/ (( A F B)) = _V ) )

Syntax hints:   <-> wb 189   \/ wo 374   = wceq 1455   _Vcvv 3057   (/)c0 3743   <.cop 3986   ` cfv 5605  (class class class)co 6320  '''cafv 38750   ((caov 38751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-res 4868  df-iota 5569  df-fun 5607  df-fv 5613  df-ov 6323  df-dfat 38752  df-afv 38753  df-aov 38754

This theorem is referenced by: (None)