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Theorem sspr 4339
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 3740 . . . . 5 (∅ ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ ∅)
2 un0 3944 . . . . 5 ({𝐵, 𝐶} ∪ ∅) = {𝐵, 𝐶}
31, 2eqtri 2648 . . . 4 (∅ ∪ {𝐵, 𝐶}) = {𝐵, 𝐶}
43sseq2i 3614 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 ⊆ {𝐵, 𝐶})
5 0ss 3949 . . . 4 ∅ ⊆ 𝐴
65biantrur 527 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
74, 6bitr3i 266 . 2 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
8 ssunpr 4338 . 2 ((∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})) ↔ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))))
9 uncom 3740 . . . . . 6 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
10 un0 3944 . . . . . 6 ({𝐵} ∪ ∅) = {𝐵}
119, 10eqtri 2648 . . . . 5 (∅ ∪ {𝐵}) = {𝐵}
1211eqeq2i 2638 . . . 4 (𝐴 = (∅ ∪ {𝐵}) ↔ 𝐴 = {𝐵})
1312orbi2i 541 . . 3 ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
14 uncom 3740 . . . . . 6 (∅ ∪ {𝐶}) = ({𝐶} ∪ ∅)
15 un0 3944 . . . . . 6 ({𝐶} ∪ ∅) = {𝐶}
1614, 15eqtri 2648 . . . . 5 (∅ ∪ {𝐶}) = {𝐶}
1716eqeq2i 2638 . . . 4 (𝐴 = (∅ ∪ {𝐶}) ↔ 𝐴 = {𝐶})
183eqeq2i 2638 . . . 4 (𝐴 = (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶})
1917, 18orbi12i 543 . . 3 ((𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))
2013, 19orbi12i 543 . 2 (((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))) ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
217, 8, 203bitri 286 1 (𝐴 ⊆ {𝐵, 𝐶} ↔ ((𝐴 = ∅ ∨ 𝐴 = {𝐵}) ∨ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wa 384   = wceq 1480  cun 3558  wss 3560  c0 3896  {csn 4153  {cpr 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-sn 4154  df-pr 4156
This theorem is referenced by:  sstp  4340  pwpr  4403  propssopi  4936  indistopon  20710
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