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

Proof of Theorem sspr

Step	Hyp	Ref	Expression
1		uncom 3757	. . . . 5 ⊢ (∅ ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ ∅)
2		un0 3967	. . . . 5 ⊢ ({𝐵, 𝐶} ∪ ∅) = {𝐵, 𝐶}
3	1, 2	eqtri 2644	. . . 4 ⊢ (∅ ∪ {𝐵, 𝐶}) = {𝐵, 𝐶}
4	3	sseq2i 3630	. . 3 ⊢ (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 ⊆ {𝐵, 𝐶})
5		0ss 3972	. . . 4 ⊢ ∅ ⊆ 𝐴
6	5	biantrur 527	. . 3 ⊢ (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
7	4, 6	bitr3i 266	. 2 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
8		ssunpr 4365	. 2 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})) ↔ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))))
9		uncom 3757	. . . . . 6 ⊢ (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
10		un0 3967	. . . . . 6 ⊢ ({𝐵} ∪ ∅) = {𝐵}
11	9, 10	eqtri 2644	. . . . 5 ⊢ (∅ ∪ {𝐵}) = {𝐵}
12	11	eqeq2i 2634	. . . 4 ⊢ (𝐴 = (∅ ∪ {𝐵}) ↔ 𝐴 = {𝐵})
13	12	orbi2i 541	. . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
14		uncom 3757	. . . . . 6 ⊢ (∅ ∪ {𝐶}) = ({𝐶} ∪ ∅)
15		un0 3967	. . . . . 6 ⊢ ({𝐶} ∪ ∅) = {𝐶}
16	14, 15	eqtri 2644	. . . . 5 ⊢ (∅ ∪ {𝐶}) = {𝐶}
17	16	eqeq2i 2634	. . . 4 ⊢ (𝐴 = (∅ ∪ {𝐶}) ↔ 𝐴 = {𝐶})
18	3	eqeq2i 2634	. . . 4 ⊢ (𝐴 = (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶})
19	17, 18	orbi12i 543	. . 3 ⊢ ((𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))
20	13, 19	orbi12i 543	. 2 ⊢ (((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
21	7, 8, 20	3bitri 286	1 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))

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