Theorem ssunpr 4794

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 4587	. . . . . 6 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
2	1	uneq2i 4112	. . . . 5 ⊢ (𝐵 ∪ {𝐶, 𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
3		unass 4118	. . . . 5 ⊢ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
4	2, 3	eqtr4i 2786	. . . 4 ⊢ (𝐵 ∪ {𝐶, 𝐷}) = ((𝐵 ∪ {𝐶}) ∪ {𝐷})
5	4	sseq2i 3960	. . 3 ⊢ (𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))
6	5	anbi2i 635	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})))
7		ssunsn2 4788	. 2 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))))
8		ssunsn 4789	. . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})))
9		un23 4120	. . . . . 6 ⊢ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = ((𝐵 ∪ {𝐷}) ∪ {𝐶})
10	9	sseq2i 3960	. . . . 5 ⊢ (𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶}))
11	10	anbi2i 635	. . . 4 ⊢ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
12		ssunsn 4789	. . . 4 ⊢ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
13	4, 9	eqtr2i 2784	. . . . . 6 ⊢ ((𝐵 ∪ {𝐷}) ∪ {𝐶}) = (𝐵 ∪ {𝐶, 𝐷})
14	13	eqeq2i 2773	. . . . 5 ⊢ (𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶}) ↔ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))
15	14	orbi2i 926	. . . 4 ⊢ ((𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
16	11, 12, 15	3bitri 300	. . 3 ⊢ (((𝐵 ∪ {𝐷}) ⊆ 𝐴 ∧ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
17	8, 16	orbi12i 928	. 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 300	1 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵 ∨ 𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∪ cun 3897   ⊆ wss 3899  {csn 4584  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-sn 4585  df-pr 4587

This theorem is used by:  sspr  4795  sstp  4796