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Theorem ssunsn 4724
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 4723 . 2 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ ((𝐵𝐴𝐴𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶}))))
2 ancom 464 . . . 4 ((𝐵𝐴𝐴𝐵) ↔ (𝐴𝐵𝐵𝐴))
3 eqss 3933 . . . 4 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
42, 3bitr4i 281 . . 3 ((𝐵𝐴𝐴𝐵) ↔ 𝐴 = 𝐵)
5 ancom 464 . . . 4 (((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
6 eqss 3933 . . . 4 (𝐴 = (𝐵 ∪ {𝐶}) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
75, 6bitr4i 281 . . 3 (((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ 𝐴 = (𝐵 ∪ {𝐶}))
84, 7orbi12i 912 . 2 (((𝐵𝐴𝐴𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴𝐴 ⊆ (𝐵 ∪ {𝐶}))) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))
91, 8bitri 278 1 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844   = wceq 1538  cun 3882  wss 3884  {csn 4528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529
This theorem is referenced by:  ssunpr  4728
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