Theorem ssunpr 4762

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

Proof of Theorem ssunpr

Step	Hyp	Ref	Expression
1		df-pr 4561	. . . . . 6 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
2	1	uneq2i 4090	. . . . 5 ⊢ (𝐵 ∪ {𝐶, 𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
3		unass 4096	. . . . 5 ⊢ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
4	2, 3	eqtr4i 2769	. . . 4 ⊢ (𝐵 ∪ {𝐶, 𝐷}) = ((𝐵 ∪ {𝐶}) ∪ {𝐷})
5	4	sseq2i 3946	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))
6	5	anbi2i 622	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})))
7		ssunsn2 4757	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))))
8		ssunsn 4758	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))
9		un23 4098	. . . . . 6 ⊢ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = ((𝐵 ∪ {𝐷}) ∪ {𝐶})
10	9	sseq2i 3946	. . . . 5 ⊢ (𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶}))
11	10	anbi2i 622	. . . 4 ⊢ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
12		ssunsn 4758	. . . 4 ⊢ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
13	4, 9	eqtr2i 2767	. . . . . 6 ⊢ ((𝐵 ∪ {𝐷}) ∪ {𝐶}) = (𝐵 ∪ {𝐶, 𝐷})
14	13	eqeq2i 2751	. . . . 5 ⊢ (𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶}) ↔ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))
15	14	orbi2i 909	. . . 4 ⊢ ((𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
16	11, 12, 15	3bitri 296	. . 3 ⊢ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
17	8, 16	orbi12i 911	. 2 ⊢ (((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))
18	6, 7, 17	3bitri 296	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539   ∪ cun 3881   ⊆ wss 3883  {csn 4558  {cpr 4560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-sn 4559  df-pr 4561

This theorem is referenced by:  sspr  4763  sstp  4764