Theorem sspr 3869

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 ⊆ {B, C} ↔ ((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})))

Proof of Theorem sspr

Step	Hyp	Ref	Expression
1		uncom 3408	. . . . 5 ⊢ (∅ ∪ {B, C}) = ({B, C} ∪ ∅)
2		un0 3575	. . . . 5 ⊢ ({B, C} ∪ ∅) = {B, C}
3	1, 2	eqtri 2373	. . . 4 ⊢ (∅ ∪ {B, C}) = {B, C}
4	3	sseq2i 3296	. . 3 ⊢ (A ⊆ (∅ ∪ {B, C}) ↔ A ⊆ {B, C})
5		0ss 3579	. . . 4 ⊢ ∅ ⊆ A
6	5	biantrur 492	. . 3 ⊢ (A ⊆ (∅ ∪ {B, C}) ↔ (∅ ⊆ A ∧ A ⊆ (∅ ∪ {B, C})))
7	4, 6	bitr3i 242	. 2 ⊢ (A ⊆ {B, C} ↔ (∅ ⊆ A ∧ A ⊆ (∅ ∪ {B, C})))
8		ssunpr 3868	. 2 ⊢ ((∅ ⊆ A ∧ A ⊆ (∅ ∪ {B, C})) ↔ ((A = ∅ ∨ A = (∅ ∪ {B})) ∨ (A = (∅ ∪ {C}) ∨ A = (∅ ∪ {B, C}))))
9		uncom 3408	. . . . . 6 ⊢ (∅ ∪ {B}) = ({B} ∪ ∅)
10		un0 3575	. . . . . 6 ⊢ ({B} ∪ ∅) = {B}
11	9, 10	eqtri 2373	. . . . 5 ⊢ (∅ ∪ {B}) = {B}
12	11	eqeq2i 2363	. . . 4 ⊢ (A = (∅ ∪ {B}) ↔ A = {B})
13	12	orbi2i 505	. . 3 ⊢ ((A = ∅ ∨ A = (∅ ∪ {B})) ↔ (A = ∅ ∨ A = {B}))
14		uncom 3408	. . . . . 6 ⊢ (∅ ∪ {C}) = ({C} ∪ ∅)
15		un0 3575	. . . . . 6 ⊢ ({C} ∪ ∅) = {C}
16	14, 15	eqtri 2373	. . . . 5 ⊢ (∅ ∪ {C}) = {C}
17	16	eqeq2i 2363	. . . 4 ⊢ (A = (∅ ∪ {C}) ↔ A = {C})
18	3	eqeq2i 2363	. . . 4 ⊢ (A = (∅ ∪ {B, C}) ↔ A = {B, C})
19	17, 18	orbi12i 507	. . 3 ⊢ ((A = (∅ ∪ {C}) ∨ A = (∅ ∪ {B, C})) ↔ (A = {C} ∨ A = {B, C}))
20	13, 19	orbi12i 507	. 2 ⊢ (((A = ∅ ∨ A = (∅ ∪ {B})) ∨ (A = (∅ ∪ {C}) ∨ A = (∅ ∪ {B, C}))) ↔ ((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})))
21	7, 8, 20	3bitri 262	1 ⊢ (A ⊆ {B, C} ↔ ((A = ∅ ∨ A = {B}) ∨ (A = {C} ∨ A = {B, C})))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  ∅c0 3550  {csn 3737  {cpr 3738

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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742

This theorem is referenced by:  sstp  3870  pwpr  3883