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Theorem sstp 4399
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
StepHypRef Expression
1 df-tp 4215 . . 3 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
21sseq2i 3663 . 2 (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))
3 0ss 4005 . . 3 ∅ ⊆ 𝐴
43biantrur 526 . 2 (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})))
5 ssunsn2 4391 . . 3 ((∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))))
63biantrur 526 . . . . 5 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}))
7 sspr 4398 . . . . 5 (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
86, 7bitr3i 266 . . . 4 ((∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
9 uncom 3790 . . . . . . . 8 (∅ ∪ {𝐷}) = ({𝐷} ∪ ∅)
10 un0 4000 . . . . . . . 8 ({𝐷} ∪ ∅) = {𝐷}
119, 10eqtri 2673 . . . . . . 7 (∅ ∪ {𝐷}) = {𝐷}
1211sseq1i 3662 . . . . . 6 ((∅ ∪ {𝐷}) ⊆ 𝐴 ↔ {𝐷} ⊆ 𝐴)
13 uncom 3790 . . . . . . 7 ({𝐵, 𝐶} ∪ {𝐷}) = ({𝐷} ∪ {𝐵, 𝐶})
1413sseq2i 3663 . . . . . 6 (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶}))
1512, 14anbi12i 733 . . . . 5 (((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ({𝐷} ⊆ 𝐴𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})))
16 ssunpr 4397 . . . . 5 (({𝐷} ⊆ 𝐴𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ∨ (𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶}))))
17 uncom 3790 . . . . . . . . 9 ({𝐷} ∪ {𝐵}) = ({𝐵} ∪ {𝐷})
18 df-pr 4213 . . . . . . . . 9 {𝐵, 𝐷} = ({𝐵} ∪ {𝐷})
1917, 18eqtr4i 2676 . . . . . . . 8 ({𝐷} ∪ {𝐵}) = {𝐵, 𝐷}
2019eqeq2i 2663 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐵}) ↔ 𝐴 = {𝐵, 𝐷})
2120orbi2i 540 . . . . . 6 ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ↔ (𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}))
22 uncom 3790 . . . . . . . . 9 ({𝐷} ∪ {𝐶}) = ({𝐶} ∪ {𝐷})
23 df-pr 4213 . . . . . . . . 9 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
2422, 23eqtr4i 2676 . . . . . . . 8 ({𝐷} ∪ {𝐶}) = {𝐶, 𝐷}
2524eqeq2i 2663 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐶}) ↔ 𝐴 = {𝐶, 𝐷})
261, 13eqtr2i 2674 . . . . . . . 8 ({𝐷} ∪ {𝐵, 𝐶}) = {𝐵, 𝐶, 𝐷}
2726eqeq2i 2663 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶, 𝐷})
2825, 27orbi12i 542 . . . . . 6 ((𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))
2921, 28orbi12i 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})))
3015, 16, 293bitri 286 . . . 4 (((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})))
318, 30orbi12i 542 . . 3 (((∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))) ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))
325, 31bitri 264 . 2 ((∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))))
332, 4, 323bitri 286 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}))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 382  wa 383   = wceq 1523  cun 3605  wss 3607  c0 3948  {csn 4210  {cpr 4212  {ctp 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-tp 4215
This theorem is referenced by:  pwtp  4463
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