Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspr Structured version   Visualization version   GIF version

Theorem sspr 4726
 Description: The subsets of a pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
sspr (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))

Proof of Theorem sspr
StepHypRef Expression
1 uncom 4080 . . . . 5 (∅ ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ ∅)
2 un0 4298 . . . . 5 ({𝐵, 𝐶} ∪ ∅) = {𝐵, 𝐶}
31, 2eqtri 2821 . . . 4 (∅ ∪ {𝐵, 𝐶}) = {𝐵, 𝐶}
43sseq2i 3944 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 ⊆ {𝐵, 𝐶})
5 0ss 4304 . . . 4 ∅ ⊆ 𝐴
65biantrur 534 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
74, 6bitr3i 280 . 2 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
8 ssunpr 4725 . 2 ((∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})) ↔ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))))
9 uncom 4080 . . . . . 6 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
10 un0 4298 . . . . . 6 ({𝐵} ∪ ∅) = {𝐵}
119, 10eqtri 2821 . . . . 5 (∅ ∪ {𝐵}) = {𝐵}
1211eqeq2i 2811 . . . 4 (𝐴 = (∅ ∪ {𝐵}) ↔ 𝐴 = {𝐵})
1312orbi2i 910 . . 3 ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
14 uncom 4080 . . . . . 6 (∅ ∪ {𝐶}) = ({𝐶} ∪ ∅)
15 un0 4298 . . . . . 6 ({𝐶} ∪ ∅) = {𝐶}
1614, 15eqtri 2821 . . . . 5 (∅ ∪ {𝐶}) = {𝐶}
1716eqeq2i 2811 . . . 4 (𝐴 = (∅ ∪ {𝐶}) ↔ 𝐴 = {𝐶})
183eqeq2i 2811 . . . 4 (𝐴 = (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶})
1917, 18orbi12i 912 . . 3 ((𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))
2013, 19orbi12i 912 . 2 (((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
217, 8, 203bitri 300 1 (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∪ cun 3879   ⊆ wss 3881  ∅c0 4243  {csn 4525  {cpr 4527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528 This theorem is referenced by:  sstp  4727  pwpr  4794  propssopi  5363  indistopon  21613  bj-prmoore  34546
 Copyright terms: Public domain W3C validator