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Theorem sstp 4335
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 4153 . . 3 {𝐵, 𝐶, 𝐷} = ({𝐵, 𝐶} ∪ {𝐷})
21sseq2i 3609 . 2 (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ 𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))
3 0ss 3944 . . 3 ∅ ⊆ 𝐴
43biantrur 527 . 2 (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})))
5 ssunsn2 4327 . . 3 ((∅ ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ((∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}) ∨ ((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}))))
63biantrur 527 . . . . 5 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ {𝐵, 𝐶}))
7 sspr 4334 . . . . 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 3735 . . . . . . . 8 (∅ ∪ {𝐷}) = ({𝐷} ∪ ∅)
10 un0 3939 . . . . . . . 8 ({𝐷} ∪ ∅) = {𝐷}
119, 10eqtri 2643 . . . . . . 7 (∅ ∪ {𝐷}) = {𝐷}
1211sseq1i 3608 . . . . . 6 ((∅ ∪ {𝐷}) ⊆ 𝐴 ↔ {𝐷} ⊆ 𝐴)
13 uncom 3735 . . . . . . 7 ({𝐵, 𝐶} ∪ {𝐷}) = ({𝐷} ∪ {𝐵, 𝐶})
1413sseq2i 3609 . . . . . 6 (𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷}) ↔ 𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶}))
1512, 14anbi12i 732 . . . . 5 (((∅ ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ({𝐵, 𝐶} ∪ {𝐷})) ↔ ({𝐷} ⊆ 𝐴𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})))
16 ssunpr 4333 . . . . 5 (({𝐷} ⊆ 𝐴𝐴 ⊆ ({𝐷} ∪ {𝐵, 𝐶})) ↔ ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ∨ (𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶}))))
17 uncom 3735 . . . . . . . . 9 ({𝐷} ∪ {𝐵}) = ({𝐵} ∪ {𝐷})
18 df-pr 4151 . . . . . . . . 9 {𝐵, 𝐷} = ({𝐵} ∪ {𝐷})
1917, 18eqtr4i 2646 . . . . . . . 8 ({𝐷} ∪ {𝐵}) = {𝐵, 𝐷}
2019eqeq2i 2633 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐵}) ↔ 𝐴 = {𝐵, 𝐷})
2120orbi2i 541 . . . . . 6 ((𝐴 = {𝐷} ∨ 𝐴 = ({𝐷} ∪ {𝐵})) ↔ (𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}))
22 uncom 3735 . . . . . . . . 9 ({𝐷} ∪ {𝐶}) = ({𝐶} ∪ {𝐷})
23 df-pr 4151 . . . . . . . . 9 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
2422, 23eqtr4i 2646 . . . . . . . 8 ({𝐷} ∪ {𝐶}) = {𝐶, 𝐷}
2524eqeq2i 2633 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐶}) ↔ 𝐴 = {𝐶, 𝐷})
261, 13eqtr2i 2644 . . . . . . . 8 ({𝐷} ∪ {𝐵, 𝐶}) = {𝐵, 𝐶, 𝐷}
2726eqeq2i 2633 . . . . . . 7 (𝐴 = ({𝐷} ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶, 𝐷})
2825, 27orbi12i 543 . . . . . 6 ((𝐴 = ({𝐷} ∪ {𝐶}) ∨ 𝐴 = ({𝐷} ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))
2921, 28orbi12i 543 . . . . 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 543 . . 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 383  wa 384   = wceq 1480  cun 3553  wss 3555  c0 3891  {csn 4148  {cpr 4150  {ctp 4152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151  df-tp 4153
This theorem is referenced by:  pwtp  4399
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