Theorem sstp 4367

Description: The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
sstp	|- ( A C_ { B , C , D } <-> ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) )

Proof of Theorem sstp

Step	Hyp	Ref	Expression
1		df-tp 4182	. . 3 |- { B , C , D } = ( { B , C } u. { D } )
2	1	sseq2i 3630	. 2 |- ( A C_ { B , C , D } <-> A C_ ( { B , C } u. { D } ) )
3		0ss 3972	. . 3 |- (/) C_ A
4	3	biantrur 527	. 2 |- ( A C_ ( { B , C } u. { D } ) <-> ( (/) C_ A /\ A C_ ( { B , C } u. { D } ) ) )
5		ssunsn2 4359	. . 3 |- ( ( (/) C_ A /\ A C_ ( { B , C } u. { D } ) ) <-> ( ( (/) C_ A /\ A C_ { B , C } ) \/ ( ( (/) u. { D } ) C_ A /\ A C_ ( { B , C } u. { D } ) ) ) )
6	3	biantrur 527	. . . . 5 |- ( A C_ { B , C } <-> ( (/) C_ A /\ A C_ { B , C } ) )
7		sspr 4366	. . . . 5 |- ( A C_ { B , C } <-> ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) )
8	6, 7	bitr3i 266	. . . 4 |- ( ( (/) C_ A /\ A C_ { B , C } ) <-> ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) )
9		uncom 3757	. . . . . . . 8 |- ( (/) u. { D } ) = ( { D } u. (/) )
10		un0 3967	. . . . . . . 8 |- ( { D } u. (/) ) = { D }
11	9, 10	eqtri 2644	. . . . . . 7 |- ( (/) u. { D } ) = { D }
12	11	sseq1i 3629	. . . . . 6 |- ( ( (/) u. { D } ) C_ A <-> { D } C_ A )
13		uncom 3757	. . . . . . 7 |- ( { B , C } u. { D } ) = ( { D } u. { B , C } )
14	13	sseq2i 3630	. . . . . 6 |- ( A C_ ( { B , C } u. { D } ) <-> A C_ ( { D } u. { B , C } ) )
15	12, 14	anbi12i 733	. . . . 5 |- ( ( ( (/) u. { D } ) C_ A /\ A C_ ( { B , C } u. { D } ) ) <-> ( { D } C_ A /\ A C_ ( { D } u. { B , C } ) ) )
16		ssunpr 4365	. . . . 5 |- ( ( { D } C_ A /\ A C_ ( { D } u. { B , C } ) ) <-> ( ( A = { D } \/ A = ( { D } u. { B } ) ) \/ ( A = ( { D } u. { C } ) \/ A = ( { D } u. { B , C } ) ) ) )
17		uncom 3757	. . . . . . . . 9 |- ( { D } u. { B } ) = ( { B } u. { D } )
18		df-pr 4180	. . . . . . . . 9 |- { B , D } = ( { B } u. { D } )
19	17, 18	eqtr4i 2647	. . . . . . . 8 |- ( { D } u. { B } ) = { B , D }
20	19	eqeq2i 2634	. . . . . . 7 |- ( A = ( { D } u. { B } ) <-> A = { B , D } )
21	20	orbi2i 541	. . . . . 6 |- ( ( A = { D } \/ A = ( { D } u. { B } ) ) <-> ( A = { D } \/ A = { B , D } ) )
22		uncom 3757	. . . . . . . . 9 |- ( { D } u. { C } ) = ( { C } u. { D } )
23		df-pr 4180	. . . . . . . . 9 |- { C , D } = ( { C } u. { D } )
24	22, 23	eqtr4i 2647	. . . . . . . 8 |- ( { D } u. { C } ) = { C , D }
25	24	eqeq2i 2634	. . . . . . 7 |- ( A = ( { D } u. { C } ) <-> A = { C , D } )
26	1, 13	eqtr2i 2645	. . . . . . . 8 |- ( { D } u. { B , C } ) = { B , C , D }
27	26	eqeq2i 2634	. . . . . . 7 |- ( A = ( { D } u. { B , C } ) <-> A = { B , C , D } )
28	25, 27	orbi12i 543	. . . . . 6 |- ( ( A = ( { D } u. { C } ) \/ A = ( { D } u. { B , C } ) ) <-> ( A = { C , D } \/ A = { B , C , D } ) )
29	21, 28	orbi12i 543	. . . . 5 |- ( ( ( A = { D } \/ A = ( { D } u. { B } ) ) \/ ( A = ( { D } u. { C } ) \/ A = ( { D } u. { B , C } ) ) ) <-> ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) )
30	15, 16, 29	3bitri 286	. . . 4 |- ( ( ( (/) u. { D } ) C_ A /\ A C_ ( { B , C } u. { D } ) ) <-> ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) )
31	8, 30	orbi12i 543	. . 3 |- ( ( ( (/) C_ A /\ A C_ { B , C } ) \/ ( ( (/) u. { D } ) C_ A /\ A C_ ( { B , C } u. { D } ) ) ) <-> ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) )
32	5, 31	bitri 264	. 2 |- ( ( (/) C_ A /\ A C_ ( { B , C } u. { D } ) ) <-> ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) )
33	2, 4, 32	3bitri 286	1 |- ( A C_ { B , C , D } <-> ( ( ( A = (/) \/ A = { B } ) \/ ( A = { C } \/ A = { B , C } ) ) \/ ( ( A = { D } \/ A = { B , D } ) \/ ( A = { C , D } \/ A = { B , C , D } ) ) ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   {ctp 4181

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-3an 1039  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-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  pwtp  4431