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Theorem ssunpr 3868
 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
ssunpr ((B A A (B ∪ {C, D})) ↔ ((A = B A = (B ∪ {C})) (A = (B ∪ {D}) A = (B ∪ {C, D}))))

Proof of Theorem ssunpr
StepHypRef Expression
1 df-pr 3742 . . . . . 6 {C, D} = ({C} ∪ {D})
21uneq2i 3415 . . . . 5 (B ∪ {C, D}) = (B ∪ ({C} ∪ {D}))
3 unass 3420 . . . . 5 ((B ∪ {C}) ∪ {D}) = (B ∪ ({C} ∪ {D}))
42, 3eqtr4i 2376 . . . 4 (B ∪ {C, D}) = ((B ∪ {C}) ∪ {D})
54sseq2i 3296 . . 3 (A (B ∪ {C, D}) ↔ A ((B ∪ {C}) ∪ {D}))
65anbi2i 675 . 2 ((B A A (B ∪ {C, D})) ↔ (B A A ((B ∪ {C}) ∪ {D})))
7 ssunsn2 3865 . 2 ((B A A ((B ∪ {C}) ∪ {D})) ↔ ((B A A (B ∪ {C})) ((B ∪ {D}) A A ((B ∪ {C}) ∪ {D}))))
8 ssunsn 3866 . . 3 ((B A A (B ∪ {C})) ↔ (A = B A = (B ∪ {C})))
9 un23 3422 . . . . . 6 ((B ∪ {C}) ∪ {D}) = ((B ∪ {D}) ∪ {C})
109sseq2i 3296 . . . . 5 (A ((B ∪ {C}) ∪ {D}) ↔ A ((B ∪ {D}) ∪ {C}))
1110anbi2i 675 . . . 4 (((B ∪ {D}) A A ((B ∪ {C}) ∪ {D})) ↔ ((B ∪ {D}) A A ((B ∪ {D}) ∪ {C})))
12 ssunsn 3866 . . . 4 (((B ∪ {D}) A A ((B ∪ {D}) ∪ {C})) ↔ (A = (B ∪ {D}) A = ((B ∪ {D}) ∪ {C})))
134, 9eqtr2i 2374 . . . . . 6 ((B ∪ {D}) ∪ {C}) = (B ∪ {C, D})
1413eqeq2i 2363 . . . . 5 (A = ((B ∪ {D}) ∪ {C}) ↔ A = (B ∪ {C, D}))
1514orbi2i 505 . . . 4 ((A = (B ∪ {D}) A = ((B ∪ {D}) ∪ {C})) ↔ (A = (B ∪ {D}) A = (B ∪ {C, D})))
1611, 12, 153bitri 262 . . 3 (((B ∪ {D}) A A ((B ∪ {C}) ∪ {D})) ↔ (A = (B ∪ {D}) A = (B ∪ {C, D})))
178, 16orbi12i 507 . 2 (((B A A (B ∪ {C})) ((B ∪ {D}) A A ((B ∪ {C}) ∪ {D}))) ↔ ((A = B A = (B ∪ {C})) (A = (B ∪ {D}) A = (B ∪ {C, D}))))
186, 7, 173bitri 262 1 ((B A A (B ∪ {C, D})) ↔ ((A = B A = (B ∪ {C})) (A = (B ∪ {D}) A = (B ∪ {C, D}))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  {csn 3737  {cpr 3738 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  sspr  3869  sstp  3870
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