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Theorem ssprr 3741
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 3452 . . . 4 ∅ ⊆ {𝐵, 𝐶}
2 sseq1 3170 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶}))
31, 2mpbiri 167 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶})
4 snsspr1 3726 . . . 4 {𝐵} ⊆ {𝐵, 𝐶}
5 sseq1 3170 . . . 4 (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶}))
64, 5mpbiri 167 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶})
73, 6jaoi 711 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶})
8 snsspr2 3727 . . . 4 {𝐶} ⊆ {𝐵, 𝐶}
9 sseq1 3170 . . . 4 (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶}))
108, 9mpbiri 167 . . 3 (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
11 eqimss 3201 . . 3 (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
1210, 11jaoi 711 . 2 ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶})
137, 12jaoi 711 1 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 703   = wceq 1348  wss 3121  c0 3414  {csn 3581  {cpr 3582
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pr 3588
This theorem is referenced by:  sstpr  3742  pwprss  3790
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