Theorem bj-disj2r 33013

Description: Relative version of ssdifin0 4050, allowing a biconditional, and of disj2 4024. This proof does not rely, even indirectly, on ssdifin0 4050 nor disj2 4024. (Contributed by BJ, 11-Nov-2021.)

Assertion

Ref	Expression
bj-disj2r	|- ( ( A i^i V ) C_ ( V \ B ) <-> ( ( A i^i B ) i^i V ) = (/) )

Proof of Theorem bj-disj2r

Step	Hyp	Ref	Expression
1		df-ss 3588	. . 3 |- ( ( A i^i V ) C_ ( V \ B ) <-> ( ( A i^i V ) i^i ( V \ B ) ) = ( A i^i V ) )
2		indif2 3870	. . . . 5 |- ( ( A i^i V ) i^i ( V \ B ) ) = ( ( ( A i^i V ) i^i V ) \ B )
3		inss1 3833	. . . . . . 7 |- ( ( A i^i V ) i^i V ) C_ ( A i^i V )
4		ssid 3624	. . . . . . . 8 |- ( A i^i V ) C_ ( A i^i V )
5		inss2 3834	. . . . . . . 8 |- ( A i^i V ) C_ V
6	4, 5	ssini 3836	. . . . . . 7 |- ( A i^i V ) C_ ( ( A i^i V ) i^i V )
7	3, 6	eqssi 3619	. . . . . 6 |- ( ( A i^i V ) i^i V ) = ( A i^i V )
8	7	difeq1i 3724	. . . . 5 |- ( ( ( A i^i V ) i^i V ) \ B ) = ( ( A i^i V ) \ B )
9	2, 8	eqtri 2644	. . . 4 |- ( ( A i^i V ) i^i ( V \ B ) ) = ( ( A i^i V ) \ B )
10	9	eqeq1i 2627	. . 3 |- ( ( ( A i^i V ) i^i ( V \ B ) ) = ( A i^i V ) <-> ( ( A i^i V ) \ B ) = ( A i^i V ) )
11		eqcom 2629	. . 3 |- ( ( ( A i^i V ) \ B ) = ( A i^i V ) <-> ( A i^i V ) = ( ( A i^i V ) \ B ) )
12	1, 10, 11	3bitri 286	. 2 |- ( ( A i^i V ) C_ ( V \ B ) <-> ( A i^i V ) = ( ( A i^i V ) \ B ) )
13		disj3 4021	. 2 |- ( ( ( A i^i V ) i^i B ) = (/) <-> ( A i^i V ) = ( ( A i^i V ) \ B ) )
14		in32 3825	. . 3 |- ( ( A i^i V ) i^i B ) = ( ( A i^i B ) i^i V )
15	14	eqeq1i 2627	. 2 |- ( ( ( A i^i V ) i^i B ) = (/) <-> ( ( A i^i B ) i^i V ) = (/) )
16	12, 13, 15	3bitr2i 288	1 |- ( ( A i^i V ) C_ ( V \ B ) <-> ( ( A i^i B ) i^i V ) = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  bj-sscon  33014