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Theorem sspr 3869
 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 3408 . . . . 5 ( ∪ {B, C}) = ({B, C} ∪ )
2 un0 3575 . . . . 5 ({B, C} ∪ ) = {B, C}
31, 2eqtri 2373 . . . 4 ( ∪ {B, C}) = {B, C}
43sseq2i 3296 . . 3 (A ( ∪ {B, C}) ↔ A {B, C})
5 0ss 3579 . . . 4 A
65biantrur 492 . . 3 (A ( ∪ {B, C}) ↔ ( A A ( ∪ {B, C})))
74, 6bitr3i 242 . 2 (A {B, C} ↔ ( A A ( ∪ {B, C})))
8 ssunpr 3868 . 2 (( A A ( ∪ {B, C})) ↔ ((A = A = ( ∪ {B})) (A = ( ∪ {C}) A = ( ∪ {B, C}))))
9 uncom 3408 . . . . . 6 ( ∪ {B}) = ({B} ∪ )
10 un0 3575 . . . . . 6 ({B} ∪ ) = {B}
119, 10eqtri 2373 . . . . 5 ( ∪ {B}) = {B}
1211eqeq2i 2363 . . . 4 (A = ( ∪ {B}) ↔ A = {B})
1312orbi2i 505 . . 3 ((A = A = ( ∪ {B})) ↔ (A = A = {B}))
14 uncom 3408 . . . . . 6 ( ∪ {C}) = ({C} ∪ )
15 un0 3575 . . . . . 6 ({C} ∪ ) = {C}
1614, 15eqtri 2373 . . . . 5 ( ∪ {C}) = {C}
1716eqeq2i 2363 . . . 4 (A = ( ∪ {C}) ↔ A = {C})
183eqeq2i 2363 . . . 4 (A = ( ∪ {B, C}) ↔ A = {B, C})
1917, 18orbi12i 507 . . 3 ((A = ( ∪ {C}) A = ( ∪ {B, C})) ↔ (A = {C} A = {B, C}))
2013, 19orbi12i 507 . 2 (((A = A = ( ∪ {B})) (A = ( ∪ {C}) A = ( ∪ {B, C}))) ↔ ((A = A = {B}) (A = {C} A = {B, C})))
217, 8, 203bitri 262 1 (A {B, C} ↔ ((A = A = {B}) (A = {C} A = {B, C})))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  ∅c0 3550  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742 This theorem is referenced by:  sstp  3870  pwpr  3883
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