Theorem sstp 4805

Description: The subsets of an unordered triple. (Contributed by Mario Carneiro, 2-Jul-2016.)

Assertion

Ref	Expression
sstp	⊢ (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))

Proof of Theorem sstp

Step	Hyp	Ref	Expression
1		df-tp 4599	. . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
2	1	sseq2i 3974	. 2 ⊢ (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))
3		0ss 4364	. . 3 ⊢ ∅ ⊆ 𝐴
4	3	biantrur 539	. 2 ⊢ (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})))
5		ssunsn2 4797	. . 3 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))))
6	3	biantrur 539	. . . . 5 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}))
7		sspr 4804	. . . . 5 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
8	6, 7	bitr3i 280	. . . 4 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
9		0un 4360	. . . . . . 7 ⊢ (∅ ∪ {𝐷}) = {𝐷}
10	9	sseq1i 3973	. . . . . 6 ⊢ ((∅ ∪ {𝐷}) ⊆ 𝐴 ↔ {𝐷} ⊆ 𝐴)
11		uncom 4120	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∪ {𝐷}) = ({𝐷} ∪ {𝐵, 𝐶})
12	11	sseq2i 3974	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶}))
13	10, 12	anbi12i 639	. . . . 5 ⊢ (((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ({𝐷} ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})))
14		ssunpr 4803	. . . . 5 ⊢ (({𝐷} ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ∨ (𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶}))))
15		uncom 4120	. . . . . . . . 9 ⊢ ({𝐷} ∪ {𝐵}) = ({𝐵} ∪ {𝐷})
16		df-pr 4597	. . . . . . . . 9 ⊢ {𝐵, 𝐷} = ({𝐵} ∪ {𝐷})
17	15, 16	eqtr4i 2795	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐵}) = {𝐵, 𝐷}
18	17	eqeq2i 2782	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐵}) ↔ 𝐴 = {𝐵, 𝐷})
19	18	orbi2i 925	. . . . . 6 ⊢ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ↔ (𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}))
20		uncom 4120	. . . . . . . . 9 ⊢ ({𝐷} ∪ {𝐶}) = ({𝐶} ∪ {𝐷})
21		df-pr 4597	. . . . . . . . 9 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
22	20, 21	eqtr4i 2795	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐶}) = {𝐶, 𝐷}
23	22	eqeq2i 2782	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐶}) ↔ 𝐴 = {𝐶, 𝐷})
24	1, 11	eqtr2i 2793	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐵, 𝐶}) = {𝐵, 𝐶, 𝐷}
25	24	eqeq2i 2782	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶, 𝐷})
26	23, 25	orbi12i 927	. . . . . 6 ⊢ ((𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))
27	19, 26	orbi12i 927	. . . . 5 ⊢ (((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ∨ (𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶}))) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})))
28	13, 14, 27	3bitri 300	. . . 4 ⊢ (((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})))
29	8, 28	orbi12i 927	. . 3 ⊢ (((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))) ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))
30	5, 29	bitri 278	. 2 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))
31	2, 4, 30	3bitri 300	1 ⊢ (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  {csn 4594  {cpr 4596  {ctp 4598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-sn 4595  df-pr 4597  df-tp 4599

This theorem is referenced by:  pwtp  4871