Theorem sstp 4307

Description: The subsets of a 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 4130	. . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
2	1	sseq2i 3593	. 2 ⊢ (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))
3		0ss 3924	. . 3 ⊢ ∅ ⊆ 𝐴
4	3	biantrur 526	. 2 ⊢ (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})))
5		ssunsn2 4299	. . 3 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))))
6	3	biantrur 526	. . . . 5 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}))
7		sspr 4306	. . . . 5 ⊢ (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
8	6, 7	bitr3i 265	. . . 4 ⊢ ((∅ ⊆ 𝐴 ∧ 𝐴 ⊆ {𝐵, 𝐶}) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
9		uncom 3719	. . . . . . . 8 ⊢ (∅ ∪ {𝐷}) = ({𝐷} ∪ ∅)
10		un0 3919	. . . . . . . 8 ⊢ ({𝐷} ∪ ∅) = {𝐷}
11	9, 10	eqtri 2632	. . . . . . 7 ⊢ (∅ ∪ {𝐷}) = {𝐷}
12	11	sseq1i 3592	. . . . . 6 ⊢ ((∅ ∪ {𝐷}) ⊆ 𝐴 ↔ {𝐷} ⊆ 𝐴)
13		uncom 3719	. . . . . . 7 ⊢ ({𝐵, 𝐶} ∪ {𝐷}) = ({𝐷} ∪ {𝐵, 𝐶})
14	13	sseq2i 3593	. . . . . 6 ⊢ (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶}))
15	12, 14	anbi12i 729	. . . . 5 ⊢ (((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ({𝐷} ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})))
16		ssunpr 4305	. . . . 5 ⊢ (({𝐷} ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ∨ (𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶}))))
17		uncom 3719	. . . . . . . . 9 ⊢ ({𝐷} ∪ {𝐵}) = ({𝐵} ∪ {𝐷})
18		df-pr 4128	. . . . . . . . 9 ⊢ {𝐵, 𝐷} = ({𝐵} ∪ {𝐷})
19	17, 18	eqtr4i 2635	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐵}) = {𝐵, 𝐷}
20	19	eqeq2i 2622	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐵}) ↔ 𝐴 = {𝐵, 𝐷})
21	20	orbi2i 540	. . . . . 6 ⊢ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ↔ (𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}))
22		uncom 3719	. . . . . . . . 9 ⊢ ({𝐷} ∪ {𝐶}) = ({𝐶} ∪ {𝐷})
23		df-pr 4128	. . . . . . . . 9 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
24	22, 23	eqtr4i 2635	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐶}) = {𝐶, 𝐷}
25	24	eqeq2i 2622	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐶}) ↔ 𝐴 = {𝐶, 𝐷})
26	1, 13	eqtr2i 2633	. . . . . . . 8 ⊢ ({𝐷} ∪ {𝐵, 𝐶}) = {𝐵, 𝐶, 𝐷}
27	26	eqeq2i 2622	. . . . . . 7 ⊢ (𝐴 = ({𝐷} ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶, 𝐷})
28	25, 27	orbi12i 542	. . . . . 6 ⊢ ((𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))
29	21, 28	orbi12i 542	. . . . 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 285	. . . 4 ⊢ (((∅ ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})))
31	8, 30	orbi12i 542	. . 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 263	. 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 285	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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  {ctp 4129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-tp 4130

This theorem is referenced by:  pwtp  4369