Theorem ssunsn 4795

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 4794	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))))
2		ancom 465	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3		eqss 3960	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	2, 3	bitr4i 281	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 = 𝐵)
5		ancom 465	. . . 4 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
6		eqss 3960	. . . 4 ⊢ (𝐴 = (𝐵 ∪ {𝐶}) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
7	5, 6	bitr4i 281	. . 3 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ 𝐴 = (𝐵 ∪ {𝐶}))
8	4, 7	orbi12i 927	. 2 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))
9	1, 8	bitri 278	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∪ cun 3911   ⊆ wss 3913  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-sn 4592

This theorem is referenced by:  ssunpr  4800