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Theorem ssunsn 3867
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 3866 . 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 3288 . . . 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 3288 . . . 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 3208   wss 3258  {csn 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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742
This theorem is referenced by:  ssunpr  3869
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