Theorem sspr 3870

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

Assertion

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

Proof of Theorem sspr

Step	Hyp	Ref	Expression
1		uncom 3409	. . . . 5 |- ((/) u. {B, C}) = ({B, C} u. (/))
2		un0 3576	. . . . 5 |- ({B, C} u. (/)) = {B, C}
3	1, 2	eqtri 2373	. . . 4 |- ((/) u. {B, C}) = {B, C}
4	3	sseq2i 3297	. . 3 |- (A C_ ((/) u. {B, C}) <-> A C_ {B, C})
5		0ss 3580	. . . 4 |- (/) C_ A
6	5	biantrur 492	. . 3 |- (A C_ ((/) u. {B, C}) <-> ((/) C_ A /\ A C_ ((/) u. {B, C})))
7	4, 6	bitr3i 242	. 2 |- (A C_ {B, C} <-> ((/) C_ A /\ A C_ ((/) u. {B, C})))
8		ssunpr 3869	. 2 |- (((/) C_ A /\ A C_ ((/) u. {B, C})) <-> ((A = (/) \/ A = ((/) u. {B})) \/ (A = ((/) u. {C}) \/ A = ((/) u. {B, C}))))
9		uncom 3409	. . . . . 6 |- ((/) u. {B}) = ({B} u. (/))
10		un0 3576	. . . . . 6 |- ({B} u. (/)) = {B}
11	9, 10	eqtri 2373	. . . . 5 |- ((/) u. {B}) = {B}
12	11	eqeq2i 2363	. . . 4 |- (A = ((/) u. {B}) <-> A = {B})
13	12	orbi2i 505	. . 3 |- ((A = (/) \/ A = ((/) u. {B})) <-> (A = (/) \/ A = {B}))
14		uncom 3409	. . . . . 6 |- ((/) u. {C}) = ({C} u. (/))
15		un0 3576	. . . . . 6 |- ({C} u. (/)) = {C}
16	14, 15	eqtri 2373	. . . . 5 |- ((/) u. {C}) = {C}
17	16	eqeq2i 2363	. . . 4 |- (A = ((/) u. {C}) <-> A = {C})
18	3	eqeq2i 2363	. . . 4 |- (A = ((/) u. {B, C}) <-> A = {B, C})
19	17, 18	orbi12i 507	. . 3 |- ((A = ((/) u. {C}) \/ A = ((/) u. {B, C})) <-> (A = {C} \/ A = {B, C}))
20	13, 19	orbi12i 507	. 2 |- (((A = (/) \/ A = ((/) u. {B})) \/ (A = ((/) u. {C}) \/ A = ((/) u. {B, C}))) <-> ((A = (/) \/ A = {B}) \/ (A = {C} \/ A = {B, C})))
21	7, 8, 20	3bitri 262	1 |- (A C_ {B, C} <-> ((A = (/) \/ A = {B}) \/ (A = {C} \/ A = {B, C})))

Syntax hints:   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   u. cun 3208   C_ wss 3258  (/)c0 3551  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743

This theorem is referenced by:  sstp  3871  pwpr  3884