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Theorem ssprr 3518
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 3249 . . . 4 ∅ ⊆ {B, 𝐶}
2 sseq1 2960 . . . 4 (A = ∅ → (A ⊆ {B, 𝐶} ↔ ∅ ⊆ {B, 𝐶}))
31, 2mpbiri 157 . . 3 (A = ∅ → A ⊆ {B, 𝐶})
4 snsspr1 3503 . . . 4 {B} ⊆ {B, 𝐶}
5 sseq1 2960 . . . 4 (A = {B} → (A ⊆ {B, 𝐶} ↔ {B} ⊆ {B, 𝐶}))
64, 5mpbiri 157 . . 3 (A = {B} → A ⊆ {B, 𝐶})
73, 6jaoi 635 . 2 ((A = ∅ A = {B}) → A ⊆ {B, 𝐶})
8 snsspr2 3504 . . . 4 {𝐶} ⊆ {B, 𝐶}
9 sseq1 2960 . . . 4 (A = {𝐶} → (A ⊆ {B, 𝐶} ↔ {𝐶} ⊆ {B, 𝐶}))
108, 9mpbiri 157 . . 3 (A = {𝐶} → A ⊆ {B, 𝐶})
11 eqimss 2991 . . 3 (A = {B, 𝐶} → A ⊆ {B, 𝐶})
1210, 11jaoi 635 . 2 ((A = {𝐶} A = {B, 𝐶}) → A ⊆ {B, 𝐶})
137, 12jaoi 635 1 (((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) → A ⊆ {B, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242  wss 2911  c0 3218  {csn 3367  {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pr 3374
This theorem is referenced by:  sstpr  3519  pwprss  3567
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