MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssunpr Structured version   Visualization version   GIF version

Theorem ssunpr 4831
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 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))

Proof of Theorem ssunpr
StepHypRef Expression
1 df-pr 4627 . . . . . 6 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
21uneq2i 4156 . . . . 5 (𝐵 ∪ {𝐶, 𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
3 unass 4162 . . . . 5 ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
42, 3eqtr4i 2758 . . . 4 (𝐵 ∪ {𝐶, 𝐷}) = ((𝐵 ∪ {𝐶}) ∪ {𝐷})
54sseq2i 4007 . . 3 (𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))
65anbi2i 622 . 2 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ (𝐵𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})))
7 ssunsn2 4826 . 2 ((𝐵𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))))
8 ssunsn 4827 . . 3 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))
9 un23 4164 . . . . . 6 ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = ((𝐵 ∪ {𝐷}) ∪ {𝐶})
109sseq2i 4007 . . . . 5 (𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶}))
1110anbi2i 622 . . . 4 (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
12 ssunsn 4827 . . . 4 (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
134, 9eqtr2i 2756 . . . . . 6 ((𝐵 ∪ {𝐷}) ∪ {𝐶}) = (𝐵 ∪ {𝐶, 𝐷})
1413eqeq2i 2740 . . . . 5 (𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶}) ↔ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))
1514orbi2i 911 . . . 4 ((𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
1611, 12, 153bitri 297 . . 3 (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
178, 16orbi12i 913 . 2 (((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))) ↔ ((𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))
186, 7, 173bitri 297 1 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ ((𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wo 846   = wceq 1534  cun 3942  wss 3944  {csn 4624  {cpr 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-sn 4625  df-pr 4627
This theorem is referenced by:  sspr  4832  sstp  4833
  Copyright terms: Public domain W3C validator