Theorem sstp 4836

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 4631	. . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
2	1	sseq2i 4013	. 2 ⊢ (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))
3		0ss 4400	. . 3 ⊢ ∅ ⊆ 𝐴
4	3	biantrur 530	. 2 ⊢ (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})))
5		ssunsn2 4827	. . 3 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))))
6	3	biantrur 530	. . . . 5 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}))
7		sspr 4835	. . . . 5 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
8	6, 7	bitr3i 277	. . . 4 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
9		uncom 4158	. . . . . . . 8 ⊢ (∅ ∪ {𝐷}) = ({𝐷} ∪ ∅)
10		un0 4394	. . . . . . . 8 ⊢ ({𝐷} ∪ ∅) = {𝐷}
11	9, 10	eqtri 2765	. . . . . . 7 ⊢ (∅ ∪ {𝐷}) = {𝐷}
12	11	sseq1i 4012	. . . . . 6 ⊢ ((∅ ∪ {𝐷}) ⊆ 𝐴 ↔ {𝐷} ⊆ 𝐴)
13		uncom 4158	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∪ {𝐷}) = ({𝐷} ∪ {𝐵, 𝐶})
14	13	sseq2i 4013	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶}))
15	12, 14	anbi12i 628	. . . . 5 ⊢ (((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ({𝐷} ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})))
16		ssunpr 4834	. . . . 5 ⊢ (({𝐷} ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ∨ (𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶}))))
17		uncom 4158	. . . . . . . . 9 ⊢ ({𝐷} ∪ {𝐵}) = ({𝐵} ∪ {𝐷})
18		df-pr 4629	. . . . . . . . 9 ⊢ {𝐵, 𝐷} = ({𝐵} ∪ {𝐷})
19	17, 18	eqtr4i 2768	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐵}) = {𝐵, 𝐷}
20	19	eqeq2i 2750	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐵}) ↔ 𝐴 = {𝐵, 𝐷})
21	20	orbi2i 913	. . . . . 6 ⊢ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ↔ (𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}))
22		uncom 4158	. . . . . . . . 9 ⊢ ({𝐷} ∪ {𝐶}) = ({𝐶} ∪ {𝐷})
23		df-pr 4629	. . . . . . . . 9 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
24	22, 23	eqtr4i 2768	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐶}) = {𝐶, 𝐷}
25	24	eqeq2i 2750	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐶}) ↔ 𝐴 = {𝐶, 𝐷})
26	1, 13	eqtr2i 2766	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐵, 𝐶}) = {𝐵, 𝐶, 𝐷}
27	26	eqeq2i 2750	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶, 𝐷})
28	25, 27	orbi12i 915	. . . . . 6 ⊢ ((𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))
29	21, 28	orbi12i 915	. . . . 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})))
30	15, 16, 29	3bitri 297	. . . 4 ⊢ (((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})))
31	8, 30	orbi12i 915	. . 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}))))
32	5, 31	bitri 275	. 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}))))
33	2, 4, 32	3bitri 297	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 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626  {cpr 4628  {ctp 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-tp 4631

This theorem is referenced by:  pwtp  4902