Theorem sspr 4193

Description: The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
sspr

Proof of Theorem sspr

Step	Hyp	Ref	Expression
1		uncom 3647	. . . . 5
2		un0 3810	. . . . 5
3	1, 2	eqtri 2486	. . . 4
4	3	sseq2i 3528	. . 3
5		0ss 3814	. . . 4
6	5	biantrur 506	. . 3
7	4, 6	bitr3i 251	. 2
8		ssunpr 4192	. 2
9		uncom 3647	. . . . . 6
10		un0 3810	. . . . . 6
11	9, 10	eqtri 2486	. . . . 5
12	11	eqeq2i 2475	. . . 4
13	12	orbi2i 519	. . 3
14		uncom 3647	. . . . . 6
15		un0 3810	. . . . . 6
16	14, 15	eqtri 2486	. . . . 5
17	16	eqeq2i 2475	. . . 4
18	3	eqeq2i 2475	. . . 4
19	17, 18	orbi12i 521	. . 3
20	13, 19	orbi12i 521	. 2
21	7, 8, 20	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

This theorem is referenced by:  sstp  4194  pwpr  4245  indistopon  19502

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