Theorem rp-fakeuninass 37862

Description: A special case where a mixture of union and intersection appears to conform to a mixed associative law. (Contributed by Richard Penner, 29-Feb-2020.)

Assertion

Ref	Expression
rp-fakeuninass	|- ( A C_ C <-> ( ( A u. B ) i^i C ) = ( A u. ( B i^i C ) ) )

Proof of Theorem rp-fakeuninass

Step	Hyp	Ref	Expression
1		rp-fakeinunass 37861	. 2 |- ( A C_ C <-> ( ( C i^i B ) u. A ) = ( C i^i ( B u. A ) ) )
2		eqcom 2629	. 2 |- ( ( ( C i^i B ) u. A ) = ( C i^i ( B u. A ) ) <-> ( C i^i ( B u. A ) ) = ( ( C i^i B ) u. A ) )
3		incom 3805	. . . 4 |- ( C i^i ( B u. A ) ) = ( ( B u. A ) i^i C )
4		uncom 3757	. . . . 5 |- ( B u. A ) = ( A u. B )
5	4	ineq1i 3810	. . . 4 |- ( ( B u. A ) i^i C ) = ( ( A u. B ) i^i C )
6	3, 5	eqtri 2644	. . 3 |- ( C i^i ( B u. A ) ) = ( ( A u. B ) i^i C )
7		uncom 3757	. . . 4 |- ( ( C i^i B ) u. A ) = ( A u. ( C i^i B ) )
8		incom 3805	. . . . 5 |- ( C i^i B ) = ( B i^i C )
9	8	uneq2i 3764	. . . 4 |- ( A u. ( C i^i B ) ) = ( A u. ( B i^i C ) )
10	7, 9	eqtri 2644	. . 3 |- ( ( C i^i B ) u. A ) = ( A u. ( B i^i C ) )
11	6, 10	eqeq12i 2636	. 2 |- ( ( C i^i ( B u. A ) ) = ( ( C i^i B ) u. A ) <-> ( ( A u. B ) i^i C ) = ( A u. ( B i^i C ) ) )
12	1, 2, 11	3bitri 286	1 |- ( A C_ C <-> ( ( A u. B ) i^i C ) = ( A u. ( B i^i C ) ) )

Syntax hints:   <-> wb 196   = wceq 1483   u. cun 3572   i^i cin 3573   C_ wss 3574

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-v 3202  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by: (None)