Theorem ssunsn 3866

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 3865	. 2 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ ((B ⊆ A ∧ A ⊆ B) ∨ ((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C}))))
2		ancom 437	. . . 4 ⊢ ((B ⊆ A ∧ A ⊆ B) ↔ (A ⊆ B ∧ B ⊆ A))
3		eqss 3287	. . . 4 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A))
4	2, 3	bitr4i 243	. . 3 ⊢ ((B ⊆ A ∧ A ⊆ B) ↔ A = B)
5		ancom 437	. . . 4 ⊢ (((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ (A ⊆ (B ∪ {C}) ∧ (B ∪ {C}) ⊆ A))
6		eqss 3287	. . . 4 ⊢ (A = (B ∪ {C}) ↔ (A ⊆ (B ∪ {C}) ∧ (B ∪ {C}) ⊆ A))
7	5, 6	bitr4i 243	. . 3 ⊢ (((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ A = (B ∪ {C}))
8	4, 7	orbi12i 507	. 2 ⊢ (((B ⊆ A ∧ A ⊆ B) ∨ ((B ∪ {C}) ⊆ A ∧ A ⊆ (B ∪ {C}))) ↔ (A = B ∨ A = (B ∪ {C})))
9	1, 8	bitri 240	1 ⊢ ((B ⊆ A ∧ A ⊆ (B ∪ {C})) ↔ (A = B ∨ A = (B ∪ {C})))

Syntax hints:   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∪ cun 3207   ⊆ wss 3257  {csn 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741

This theorem is referenced by:  ssunpr  3868