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Theorem ssprr 3653
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 3371 . . . 4 ∅ ⊆ {𝐵, 𝐶}
2 sseq1 3090 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶}))
31, 2mpbiri 167 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶})
4 snsspr1 3638 . . . 4 {𝐵} ⊆ {𝐵, 𝐶}
5 sseq1 3090 . . . 4 (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶}))
64, 5mpbiri 167 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶})
73, 6jaoi 690 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶})
8 snsspr2 3639 . . . 4 {𝐶} ⊆ {𝐵, 𝐶}
9 sseq1 3090 . . . 4 (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶}))
108, 9mpbiri 167 . . 3 (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
11 eqimss 3121 . . 3 (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
1210, 11jaoi 690 . 2 ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶})
137, 12jaoi 690 1 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 682   = wceq 1316  wss 3041  c0 3333  {csn 3497  {cpr 3498
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pr 3504
This theorem is referenced by:  sstpr  3654  pwprss  3702
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