Theorem ssunsn 4786

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

Step	Hyp	Ref	Expression
1		ssunsn2 4785	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))))
2		ancom 460	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3		eqss 3951	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	2, 3	bitr4i 278	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 = 𝐵)
5		ancom 460	. . . 4 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
6		eqss 3951	. . . 4 ⊢ (𝐴 = (𝐵 ∪ {𝐶}) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
7	5, 6	bitr4i 278	. . 3 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ 𝐴 = (𝐵 ∪ {𝐶}))
8	4, 7	orbi12i 915	. 2 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))
9	1, 8	bitri 275	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∪ cun 3901   ⊆ wss 3903  {csn 4582

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-sn 4583

This theorem is referenced by:  ssunpr  4792