Theorem sspr 4746

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 4067	. . . . 5 ⊢ (∅ ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ ∅)
2		un0 4305	. . . . 5 ⊢ ({𝐵, 𝐶} ∪ ∅) = {𝐵, 𝐶}
3	1, 2	eqtri 2765	. . . 4 ⊢ (∅ ∪ {𝐵, 𝐶}) = {𝐵, 𝐶}
4	3	sseq2i 3930	. . 3 ⊢ (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 ⊆ {𝐵, 𝐶})
5		0ss 4311	. . . 4 ⊢ ∅ ⊆ 𝐴
6	5	biantrur 534	. . 3 ⊢ (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
7	4, 6	bitr3i 280	. 2 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
8		ssunpr 4745	. 2 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})) ↔ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))))
9		uncom 4067	. . . . . 6 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
10		un0 4305	. . . . . 6 ⊢ ({𝐵} ∪ ∅) = {𝐵}
11	9, 10	eqtri 2765	. . . . 5 ⊢ (∅ ∪ {𝐵}) = {𝐵}
12	11	eqeq2i 2750	. . . 4 ⊢ (𝐴 = (∅ ∪ {𝐵}) ↔ 𝐴 = {𝐵})
13	12	orbi2i 913	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
14		uncom 4067	. . . . . 6 ⊢ (∅ ∪ {𝐶}) = ({𝐶} ∪ ∅)
15		un0 4305	. . . . . 6 ⊢ ({𝐶} ∪ ∅) = {𝐶}
16	14, 15	eqtri 2765	. . . . 5 ⊢ (∅ ∪ {𝐶}) = {𝐶}
17	16	eqeq2i 2750	. . . 4 ⊢ (𝐴 = (∅ ∪ {𝐶}) ↔ 𝐴 = {𝐶})
18	3	eqeq2i 2750	. . . 4 ⊢ (𝐴 = (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶})
19	17, 18	orbi12i 915	. . 3 ⊢ ((𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))
20	13, 19	orbi12i 915	. 2 ⊢ (((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
21	7, 8, 20	3bitri 300	1 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 847   = wceq 1543   ∪ cun 3864   ⊆ wss 3866  ∅c0 4237  {csn 4541  {cpr 4543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-sn 4542  df-pr 4544

This theorem is referenced by:  sstp  4747  pwpr  4813  propssopi  5391  indistopon  21898  bj-prmoore  35021