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Theorem ssprr 3600
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 3321 . . . 4 ∅ ⊆ {𝐵, 𝐶}
2 sseq1 3047 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶}))
31, 2mpbiri 166 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶})
4 snsspr1 3585 . . . 4 {𝐵} ⊆ {𝐵, 𝐶}
5 sseq1 3047 . . . 4 (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶}))
64, 5mpbiri 166 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶})
73, 6jaoi 671 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶})
8 snsspr2 3586 . . . 4 {𝐶} ⊆ {𝐵, 𝐶}
9 sseq1 3047 . . . 4 (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶}))
108, 9mpbiri 166 . . 3 (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
11 eqimss 3078 . . 3 (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
1210, 11jaoi 671 . 2 ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶})
137, 12jaoi 671 1 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 664   = wceq 1289  wss 2999  c0 3286  {csn 3446  {cpr 3447
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pr 3453
This theorem is referenced by:  sstpr  3601  pwprss  3649
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