Theorem ecin0 34117

Description: Two ways of saying that the coset of A and the coset of B have no elements in common. (Contributed by Peter Mazsa, 1-Dec-2018.)

Assertion

Ref	Expression
ecin0	|- ( ( A e. V /\ B e. W ) -> ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( A R x -> -. B R x ) ) )

Distinct variable groups:   x, A   x, B   x, R   x, V   x, W

Proof of Theorem ecin0

Step	Hyp	Ref	Expression
1		disj1 4019	. 2 |- ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( x e. [ A ] R -> -. x e. [ B ] R ) )
2		elecg 7785	. . . . . 6 |- ( ( x e. _V /\ A e. V ) -> ( x e. [ A ] R <-> A R x ) )
3	2	el2v1 33985	. . . . 5 |- ( A e. V -> ( x e. [ A ] R <-> A R x ) )
4	3	adantr 481	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( x e. [ A ] R <-> A R x ) )
5		elecALTV 34030	. . . . . . 7 |- ( ( B e. W /\ x e. _V ) -> ( x e. [ B ] R <-> B R x ) )
6	5	el2v2 33986	. . . . . 6 |- ( B e. W -> ( x e. [ B ] R <-> B R x ) )
7	6	adantl 482	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( x e. [ B ] R <-> B R x ) )
8	7	notbid 308	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( -. x e. [ B ] R <-> -. B R x ) )
9	4, 8	imbi12d 334	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( x e. [ A ] R -> -. x e. [ B ] R ) <-> ( A R x -> -. B R x ) ) )
10	9	albidv 1849	. 2 |- ( ( A e. V /\ B e. W ) -> ( A. x ( x e. [ A ] R -> -. x e. [ B ] R ) <-> A. x ( A R x -> -. B R x ) ) )
11	1, 10	syl5bb 272	1 |- ( ( A e. V /\ B e. W ) -> ( ( [ A ] R i^i [ B ] R ) = (/) <-> A. x ( A R x -> -. B R x ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   (/)c0 3915   class class class wbr 4653   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by:  ecinn0  34118