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	⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))

Proof of Theorem ssunsn

Step	Hyp	Ref	Expression
1		ssunsn2 4493	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))))
2		ancom 452	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
3		eqss 3767	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
4	2, 3	bitr4i 267	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ↔ 𝐴 = 𝐵)
5		ancom 452	. . . 4 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
6		eqss 3767	. . . 4 ⊢ (𝐴 = (𝐵 ∪ {𝐶}) ↔ (𝐴 ⊆ (𝐵 ∪ {𝐶}) ∧ (𝐵 ∪ {𝐶}) ⊆ 𝐴))
7	5, 6	bitr4i 267	. . 3 ⊢ (((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ 𝐴 = (𝐵 ∪ {𝐶}))
8	4, 7	orbi12i 900	. 2 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝐵) ∨ ((𝐵 ∪ {𝐶}) ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶}))) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))
9	1, 8	bitri 264	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))

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