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Theorem ssunsn 4494
 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 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))

Proof of Theorem ssunsn
StepHypRef Expression
1 ssunsn2 4493 . 2 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ ((𝐵𝐴𝐴𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶}))))
2 ancom 452 . . . 4 ((𝐵𝐴𝐴𝐵) ↔ (𝐴𝐵𝐵𝐴))
3 eqss 3767 . . . 4 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
42, 3bitr4i 267 . . 3 ((𝐵𝐴𝐴𝐵) ↔ 𝐴 = 𝐵)
5 ancom 452 . . . 4 (((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
6 eqss 3767 . . . 4 (𝐴 = (𝐵 ∪ {𝐶}) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
75, 6bitr4i 267 . . 3 (((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ 𝐴 = (𝐵 ∪ {𝐶}))
84, 7orbi12i 900 . 2 (((𝐵𝐴𝐴𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶}))) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))
91, 8bitri 264 1 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 382   ∨ wo 836   = wceq 1631   ∪ cun 3721   ⊆ wss 3723  {csn 4316 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-sn 4317 This theorem is referenced by:  ssunpr  4498
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