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

Theorem sspr 4760
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 4128 . . . . 5 (∅ ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ ∅)
2 un0 4343 . . . . 5 ({𝐵, 𝐶} ∪ ∅) = {𝐵, 𝐶}
31, 2eqtri 2844 . . . 4 (∅ ∪ {𝐵, 𝐶}) = {𝐵, 𝐶}
43sseq2i 3995 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 ⊆ {𝐵, 𝐶})
5 0ss 4349 . . . 4 ∅ ⊆ 𝐴
65biantrur 531 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
74, 6bitr3i 278 . 2 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
8 ssunpr 4759 . 2 ((∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})) ↔ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))))
9 uncom 4128 . . . . . 6 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
10 un0 4343 . . . . . 6 ({𝐵} ∪ ∅) = {𝐵}
119, 10eqtri 2844 . . . . 5 (∅ ∪ {𝐵}) = {𝐵}
1211eqeq2i 2834 . . . 4 (𝐴 = (∅ ∪ {𝐵}) ↔ 𝐴 = {𝐵})
1312orbi2i 906 . . 3 ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
14 uncom 4128 . . . . . 6 (∅ ∪ {𝐶}) = ({𝐶} ∪ ∅)
15 un0 4343 . . . . . 6 ({𝐶} ∪ ∅) = {𝐶}
1614, 15eqtri 2844 . . . . 5 (∅ ∪ {𝐶}) = {𝐶}
1716eqeq2i 2834 . . . 4 (𝐴 = (∅ ∪ {𝐶}) ↔ 𝐴 = {𝐶})
183eqeq2i 2834 . . . 4 (𝐴 = (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶})
1917, 18orbi12i 908 . . 3 ((𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))
2013, 19orbi12i 908 . 2 (((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
217, 8, 203bitri 298 1 (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wo 841   = wceq 1528  cun 3933  wss 3935  c0 4290  {csn 4559  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4560  df-pr 4562
This theorem is referenced by:  sstp  4761  pwpr  4826  propssopi  5390  indistopon  21539
  Copyright terms: Public domain W3C validator