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Theorem ssunsn 3866
 Description: Possible values for a set sandwiched between another set and it plus a singleton. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
ssunsn ((B A A (B ∪ {C})) ↔ (A = B A = (B ∪ {C})))

Proof of Theorem ssunsn
StepHypRef Expression
1 ssunsn2 3865 . 2 ((B A A (B ∪ {C})) ↔ ((B A A B) ((B ∪ {C}) A A (B ∪ {C}))))
2 ancom 437 . . . 4 ((B A A B) ↔ (A B B A))
3 eqss 3287 . . . 4 (A = B ↔ (A B B A))
42, 3bitr4i 243 . . 3 ((B A A B) ↔ A = B)
5 ancom 437 . . . 4 (((B ∪ {C}) A A (B ∪ {C})) ↔ (A (B ∪ {C}) (B ∪ {C}) A))
6 eqss 3287 . . . 4 (A = (B ∪ {C}) ↔ (A (B ∪ {C}) (B ∪ {C}) A))
75, 6bitr4i 243 . . 3 (((B ∪ {C}) A A (B ∪ {C})) ↔ A = (B ∪ {C}))
84, 7orbi12i 507 . 2 (((B A A B) ((B ∪ {C}) A A (B ∪ {C}))) ↔ (A = B A = (B ∪ {C})))
91, 8bitri 240 1 ((B A A (B ∪ {C})) ↔ (A = B A = (B ∪ {C})))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  {csn 3737 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 This theorem is referenced by:  ssunpr  3868
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