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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
StepHypRef Expression
1 df-tp 4130 . . 3 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
21sseq2i 3593 . 2 (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))
3 0ss 3924 . . 3 ∅ ⊆ 𝐴
43biantrur 526 . 2 (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})))
5 ssunsn2 4299 . . 3 ((∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))))
63biantrur 526 . . . . 5 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}))
7 sspr 4306 . . . . 5 (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
86, 7bitr3i 265 . . . 4 ((∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
9 uncom 3719 . . . . . . . 8 (∅ ∪ {𝐷}) = ({𝐷} ∪ ∅)
10 un0 3919 . . . . . . . 8 ({𝐷} ∪ ∅) = {𝐷}
119, 10eqtri 2632 . . . . . . 7 (∅ ∪ {𝐷}) = {𝐷}
1211sseq1i 3592 . . . . . 6 ((∅ ∪ {𝐷}) ⊆ 𝐴 ↔ {𝐷} ⊆ 𝐴)
13 uncom 3719 . . . . . . 7 ({𝐵, 𝐶} ∪ {𝐷}) = ({𝐷} ∪ {𝐵, 𝐶})
1413sseq2i 3593 . . . . . 6 (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶}))
1512, 14anbi12i 729 . . . . 5 (((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ({𝐷} ⊆ 𝐴𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})))
16 ssunpr 4305 . . . . 5 (({𝐷} ⊆ 𝐴𝐴 ⊆ ({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})
1917, 18eqtr4i 2635 . . . . . . . 8 ({𝐷} ∪ {𝐵}) = {𝐵, 𝐷}
2019eqeq2i 2622 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐵}) ↔ 𝐴 = {𝐵, 𝐷})
2120orbi2i 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})
2422, 23eqtr4i 2635 . . . . . . . 8 ({𝐷} ∪ {𝐶}) = {𝐶, 𝐷}
2524eqeq2i 2622 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐶}) ↔ 𝐴 = {𝐶, 𝐷})
261, 13eqtr2i 2633 . . . . . . . 8 ({𝐷} ∪ {𝐵, 𝐶}) = {𝐵, 𝐶, 𝐷}
2726eqeq2i 2622 . . . . . . 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 285 . . . 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 263 . 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 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}))))
Colors of variables: wff setvar class
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
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