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Theorem ssprr 3501
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 3232 . . . 4 ∅ ⊆ {B, 𝐶}
2 sseq1 2943 . . . 4 (A = ∅ → (A ⊆ {B, 𝐶} ↔ ∅ ⊆ {B, 𝐶}))
31, 2mpbiri 157 . . 3 (A = ∅ → A ⊆ {B, 𝐶})
4 snsspr1 3486 . . . 4 {B} ⊆ {B, 𝐶}
5 sseq1 2943 . . . 4 (A = {B} → (A ⊆ {B, 𝐶} ↔ {B} ⊆ {B, 𝐶}))
64, 5mpbiri 157 . . 3 (A = {B} → A ⊆ {B, 𝐶})
73, 6jaoi 623 . 2 ((A = ∅ A = {B}) → A ⊆ {B, 𝐶})
8 snsspr2 3487 . . . 4 {𝐶} ⊆ {B, 𝐶}
9 sseq1 2943 . . . 4 (A = {𝐶} → (A ⊆ {B, 𝐶} ↔ {𝐶} ⊆ {B, 𝐶}))
108, 9mpbiri 157 . . 3 (A = {𝐶} → A ⊆ {B, 𝐶})
11 eqimss 2974 . . 3 (A = {B, 𝐶} → A ⊆ {B, 𝐶})
1210, 11jaoi 623 . 2 ((A = {𝐶} A = {B, 𝐶}) → A ⊆ {B, 𝐶})
137, 12jaoi 623 1 (((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) → A ⊆ {B, 𝐶})
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228  wss 2894  c0 3201  {csn 3350  {cpr 3351
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pr 3357
This theorem is referenced by:  sstpr  3502  pwprss  3550
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