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Theorem ssprr 3683
Description: The subsets of a pair. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
ssprr (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})

Proof of Theorem ssprr
StepHypRef Expression
1 0ss 3401 . . . 4 ∅ ⊆ {𝐵, 𝐶}
2 sseq1 3120 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶}))
31, 2mpbiri 167 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶})
4 snsspr1 3668 . . . 4 {𝐵} ⊆ {𝐵, 𝐶}
5 sseq1 3120 . . . 4 (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶}))
64, 5mpbiri 167 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶})
73, 6jaoi 705 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶})
8 snsspr2 3669 . . . 4 {𝐶} ⊆ {𝐵, 𝐶}
9 sseq1 3120 . . . 4 (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶}))
108, 9mpbiri 167 . . 3 (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
11 eqimss 3151 . . 3 (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
1210, 11jaoi 705 . 2 ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶})
137, 12jaoi 705 1 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 697   = wceq 1331  wss 3071  c0 3363  {csn 3527  {cpr 3528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pr 3534
This theorem is referenced by:  sstpr  3684  pwprss  3732
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