Theorem sstp 4194

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

Assertion

Ref	Expression
sstp

Proof of Theorem sstp

Step	Hyp	Ref	Expression
1		df-tp 4034	. . 3
2	1	sseq2i 3528	. 2
3		0ss 3814	. . 3
4	3	biantrur 506	. 2
5		ssunsn2 4189	. . 3
6	3	biantrur 506	. . . . 5
7		sspr 4193	. . . . 5
8	6, 7	bitr3i 251	. . . 4
9		uncom 3647	. . . . . . . 8
10		un0 3810	. . . . . . . 8
11	9, 10	eqtri 2486	. . . . . . 7
12	11	sseq1i 3527	. . . . . 6
13		uncom 3647	. . . . . . 7
14	13	sseq2i 3528	. . . . . 6
15	12, 14	anbi12i 697	. . . . 5
16		ssunpr 4192	. . . . 5
17		uncom 3647	. . . . . . . . 9
18		df-pr 4032	. . . . . . . . 9
19	17, 18	eqtr4i 2489	. . . . . . . 8
20	19	eqeq2i 2475	. . . . . . 7
21	20	orbi2i 519	. . . . . 6
22		uncom 3647	. . . . . . . . 9
23		df-pr 4032	. . . . . . . . 9
24	22, 23	eqtr4i 2489	. . . . . . . 8
25	24	eqeq2i 2475	. . . . . . 7
26	1, 13	eqtr2i 2487	. . . . . . . 8
27	26	eqeq2i 2475	. . . . . . 7
28	25, 27	orbi12i 521	. . . . . 6
29	21, 28	orbi12i 521	. . . . 5
30	15, 16, 29	3bitri 271	. . . 4
31	8, 30	orbi12i 521	. . 3
32	5, 31	bitri 249	. 2
33	2, 4, 32	3bitri 271	1

Syntax hints:  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  u.cun 3473  C_wss 3475  c0 3784  {csn 4029  {cpr 4031  {ctp 4033

This theorem is referenced by:  pwtp  4246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030  df-pr 4032  df-tp 4034