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Theorem sstp 3870
 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 3743 . . 3 {B, C, D} = ({B, C} ∪ {D})
21sseq2i 3296 . 2 (A {B, C, D} ↔ A ({B, C} ∪ {D}))
3 0ss 3579 . . 3 A
43biantrur 492 . 2 (A ({B, C} ∪ {D}) ↔ ( A A ({B, C} ∪ {D})))
5 ssunsn2 3865 . . 3 (( A A ({B, C} ∪ {D})) ↔ (( A A {B, C}) (( ∪ {D}) A A ({B, C} ∪ {D}))))
63biantrur 492 . . . . 5 (A {B, C} ↔ ( A A {B, C}))
7 sspr 3869 . . . . 5 (A {B, C} ↔ ((A = A = {B}) (A = {C} A = {B, C})))
86, 7bitr3i 242 . . . 4 (( A A {B, C}) ↔ ((A = A = {B}) (A = {C} A = {B, C})))
9 uncom 3408 . . . . . . . 8 ( ∪ {D}) = ({D} ∪ )
10 un0 3575 . . . . . . . 8 ({D} ∪ ) = {D}
119, 10eqtri 2373 . . . . . . 7 ( ∪ {D}) = {D}
1211sseq1i 3295 . . . . . 6 (( ∪ {D}) A ↔ {D} A)
13 uncom 3408 . . . . . . 7 ({B, C} ∪ {D}) = ({D} ∪ {B, C})
1413sseq2i 3296 . . . . . 6 (A ({B, C} ∪ {D}) ↔ A ({D} ∪ {B, C}))
1512, 14anbi12i 678 . . . . 5 ((( ∪ {D}) A A ({B, C} ∪ {D})) ↔ ({D} A A ({D} ∪ {B, C})))
16 ssunpr 3868 . . . . 5 (({D} A A ({D} ∪ {B, C})) ↔ ((A = {D} A = ({D} ∪ {B})) (A = ({D} ∪ {C}) A = ({D} ∪ {B, C}))))
17 uncom 3408 . . . . . . . . 9 ({D} ∪ {B}) = ({B} ∪ {D})
18 df-pr 3742 . . . . . . . . 9 {B, D} = ({B} ∪ {D})
1917, 18eqtr4i 2376 . . . . . . . 8 ({D} ∪ {B}) = {B, D}
2019eqeq2i 2363 . . . . . . 7 (A = ({D} ∪ {B}) ↔ A = {B, D})
2120orbi2i 505 . . . . . 6 ((A = {D} A = ({D} ∪ {B})) ↔ (A = {D} A = {B, D}))
22 uncom 3408 . . . . . . . . 9 ({D} ∪ {C}) = ({C} ∪ {D})
23 df-pr 3742 . . . . . . . . 9 {C, D} = ({C} ∪ {D})
2422, 23eqtr4i 2376 . . . . . . . 8 ({D} ∪ {C}) = {C, D}
2524eqeq2i 2363 . . . . . . 7 (A = ({D} ∪ {C}) ↔ A = {C, D})
261, 13eqtr2i 2374 . . . . . . . 8 ({D} ∪ {B, C}) = {B, C, D}
2726eqeq2i 2363 . . . . . . 7 (A = ({D} ∪ {B, C}) ↔ A = {B, C, D})
2825, 27orbi12i 507 . . . . . 6 ((A = ({D} ∪ {C}) A = ({D} ∪ {B, C})) ↔ (A = {C, D} A = {B, C, D}))
2921, 28orbi12i 507 . . . . 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 262 . . . 4 ((( ∪ {D}) A A ({B, C} ∪ {D})) ↔ ((A = {D} A = {B, D}) (A = {C, D} A = {B, C, D})))
318, 30orbi12i 507 . . 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}))))
325, 31bitri 240 . 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}))))
332, 4, 323bitri 262 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 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  ∅c0 3550  {csn 3737  {cpr 3738  {ctp 3739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-tp 3743 This theorem is referenced by:  pwtp  3884
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