ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sstpr GIF version

Theorem sstpr 3557
Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
sstpr ((((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})

Proof of Theorem sstpr
StepHypRef Expression
1 ssprr 3556 . . 3 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
2 prsstp12 3546 . . 3 {𝐵, 𝐶} ⊆ {𝐵, 𝐶, 𝐷}
31, 2syl6ss 3012 . 2 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
4 snsstp3 3545 . . . . 5 {𝐷} ⊆ {𝐵, 𝐶, 𝐷}
5 sseq1 3021 . . . . 5 (𝐴 = {𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
64, 5mpbiri 166 . . . 4 (𝐴 = {𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
7 prsstp13 3547 . . . . 5 {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
8 sseq1 3021 . . . . 5 (𝐴 = {𝐵, 𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐵, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
97, 8mpbiri 166 . . . 4 (𝐴 = {𝐵, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
106, 9jaoi 669 . . 3 ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
11 prsstp23 3548 . . . . 5 {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}
12 sseq1 3021 . . . . 5 (𝐴 = {𝐶, 𝐷} → (𝐴 ⊆ {𝐵, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {𝐵, 𝐶, 𝐷}))
1311, 12mpbiri 166 . . . 4 (𝐴 = {𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
14 eqimss 3052 . . . 4 (𝐴 = {𝐵, 𝐶, 𝐷} → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
1513, 14jaoi 669 . . 3 ((𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
1610, 15jaoi 669 . 2 (((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷})) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
173, 16jaoi 669 1 ((((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) ∨ ((𝐴 = {𝐷} ∨ 𝐴 = {𝐵, 𝐷}) ∨ (𝐴 = {𝐶, 𝐷} ∨ 𝐴 = {𝐵, 𝐶, 𝐷}))) → 𝐴 ⊆ {𝐵, 𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1285  wss 2974  c0 3258  {csn 3406  {cpr 3407  {ctp 3408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3or 921  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-sn 3412  df-pr 3413  df-tp 3414
This theorem is referenced by:  pwtpss  3606
  Copyright terms: Public domain W3C validator