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Theorem sspr 4398
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 3790 . . . . 5 (∅ ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ ∅)
2 un0 4000 . . . . 5 ({𝐵, 𝐶} ∪ ∅) = {𝐵, 𝐶}
31, 2eqtri 2673 . . . 4 (∅ ∪ {𝐵, 𝐶}) = {𝐵, 𝐶}
43sseq2i 3663 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 ⊆ {𝐵, 𝐶})
5 0ss 4005 . . . 4 ∅ ⊆ 𝐴
65biantrur 526 . . 3 (𝐴 ⊆ (∅ ∪ {𝐵, 𝐶}) ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
74, 6bitr3i 266 . 2 (𝐴 ⊆ {𝐵, 𝐶} ↔ (∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})))
8 ssunpr 4397 . 2 ((∅ ⊆ 𝐴𝐴 ⊆ (∅ ∪ {𝐵, 𝐶})) ↔ ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ∨ (𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶}))))
9 uncom 3790 . . . . . 6 (∅ ∪ {𝐵}) = ({𝐵} ∪ ∅)
10 un0 4000 . . . . . 6 ({𝐵} ∪ ∅) = {𝐵}
119, 10eqtri 2673 . . . . 5 (∅ ∪ {𝐵}) = {𝐵}
1211eqeq2i 2663 . . . 4 (𝐴 = (∅ ∪ {𝐵}) ↔ 𝐴 = {𝐵})
1312orbi2i 540 . . 3 ((𝐴 = ∅ ∨ 𝐴 = (∅ ∪ {𝐵})) ↔ (𝐴 = ∅ ∨ 𝐴 = {𝐵}))
14 uncom 3790 . . . . . 6 (∅ ∪ {𝐶}) = ({𝐶} ∪ ∅)
15 un0 4000 . . . . . 6 ({𝐶} ∪ ∅) = {𝐶}
1614, 15eqtri 2673 . . . . 5 (∅ ∪ {𝐶}) = {𝐶}
1716eqeq2i 2663 . . . 4 (𝐴 = (∅ ∪ {𝐶}) ↔ 𝐴 = {𝐶})
183eqeq2i 2663 . . . 4 (𝐴 = (∅ ∪ {𝐵, 𝐶}) ↔ 𝐴 = {𝐵, 𝐶})
1917, 18orbi12i 542 . . 3 ((𝐴 = (∅ ∪ {𝐶}) ∨ 𝐴 = (∅ ∪ {𝐵, 𝐶})) ↔ (𝐴 = {𝐶} ∨ 𝐴 = {𝐵, 𝐶}))
2013, 19orbi12i 542 . 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 382  wa 383   = wceq 1523  cun 3605  wss 3607  c0 3948  {csn 4210  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213
This theorem is referenced by:  sstp  4399  pwpr  4462  propssopi  5000  indistopon  20853
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