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Theorem sspr 4366

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 3757	. . . . 5
2		un0 3967	. . . . 5
3	1, 2	eqtri 2644	. . . 4
4	3	sseq2i 3630	. . 3
5		0ss 3972	. . . 4
6	5	biantrur 527	. . 3 $/\$
7	4, 6	bitr3i 266	. 2 $/\$
8		ssunpr 4365	. 2 $/\$ $\/$ $\/$ $\/$
9		uncom 3757	. . . . . 6
10		un0 3967	. . . . . 6
11	9, 10	eqtri 2644	. . . . 5
12	11	eqeq2i 2634	. . . 4
13	12	orbi2i 541	. . 3 $\/$ $\/$
14		uncom 3757	. . . . . 6
15		un0 3967	. . . . . 6
16	14, 15	eqtri 2644	. . . . 5
17	16	eqeq2i 2634	. . . 4
18	3	eqeq2i 2634	. . . 4
19	17, 18	orbi12i 543	. . 3 $\/$ $\/$
20	13, 19	orbi12i 543	. 2 $\/$ $\/$ $\/$ $\/$ $\/$ $\/$
21	7, 8, 20	3bitri 286	1 $\/$ $\/$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wb 196 $\/$ wo 383 $/\$ wa 384

wceq 1483

cun 3572

wss 3574

c0 3915

csn 4177

cpr 4179

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

This theorem is referenced by: sstp 4367 pwpr 4430 propssopi 4971 indistopon 20805