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Theorem ssprr 3555
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 3283 . . . 4 ∅ ⊆ {𝐵, 𝐶}
2 sseq1 2994 . . . 4 (𝐴 = ∅ → (𝐴 ⊆ {𝐵, 𝐶} ↔ ∅ ⊆ {𝐵, 𝐶}))
31, 2mpbiri 161 . . 3 (𝐴 = ∅ → 𝐴 ⊆ {𝐵, 𝐶})
4 snsspr1 3540 . . . 4 {𝐵} ⊆ {𝐵, 𝐶}
5 sseq1 2994 . . . 4 (𝐴 = {𝐵} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐵} ⊆ {𝐵, 𝐶}))
64, 5mpbiri 161 . . 3 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵, 𝐶})
73, 6jaoi 646 . 2 ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) → 𝐴 ⊆ {𝐵, 𝐶})
8 snsspr2 3541 . . . 4 {𝐶} ⊆ {𝐵, 𝐶}
9 sseq1 2994 . . . 4 (𝐴 = {𝐶} → (𝐴 ⊆ {𝐵, 𝐶} ↔ {𝐶} ⊆ {𝐵, 𝐶}))
108, 9mpbiri 161 . . 3 (𝐴 = {𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
11 eqimss 3025 . . 3 (𝐴 = {𝐵, 𝐶} → 𝐴 ⊆ {𝐵, 𝐶})
1210, 11jaoi 646 . 2 ((𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}) → 𝐴 ⊆ {𝐵, 𝐶})
137, 12jaoi 646 1 (((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})) → 𝐴 ⊆ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wss 2945  c0 3252  {csn 3403  {cpr 3404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pr 3410
This theorem is referenced by:  sstpr  3556  pwprss  3604
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