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Theorem ssunpr 4759
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 4562 . . . . . 6 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
21uneq2i 4135 . . . . 5 (𝐵 ∪ {𝐶, 𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
3 unass 4141 . . . . 5 ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = (𝐵 ∪ ({𝐶} ∪ {𝐷}))
42, 3eqtr4i 2847 . . . 4 (𝐵 ∪ {𝐶, 𝐷}) = ((𝐵 ∪ {𝐶}) ∪ {𝐷})
54sseq2i 3995 . . 3 (𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷}) ↔ 𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))
65anbi2i 622 . 2 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶, 𝐷})) ↔ (𝐵𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})))
7 ssunsn2 4754 . 2 ((𝐵𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))))
8 ssunsn 4755 . . 3 ((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ↔ (𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})))
9 un23 4143 . . . . . 6 ((𝐵 ∪ {𝐶}) ∪ {𝐷}) = ((𝐵 ∪ {𝐷}) ∪ {𝐶})
109sseq2i 3995 . . . . 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 4755 . . . 4 (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})))
134, 9eqtr2i 2845 . . . . . 6 ((𝐵 ∪ {𝐷}) ∪ {𝐶}) = (𝐵 ∪ {𝐶, 𝐷})
1413eqeq2i 2834 . . . . 5 (𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶}) ↔ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))
1514orbi2i 906 . . . 4 ((𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = ((𝐵 ∪ {𝐷}) ∪ {𝐶})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
1611, 12, 153bitri 298 . . 3 (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷})) ↔ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷})))
178, 16orbi12i 908 . 2 (((𝐵𝐴𝐴 ⊆ (𝐵 ∪ {𝐶})) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ ((𝐵 ∪ {𝐶}) ∪ {𝐷}))) ↔ ((𝐴 = 𝐵𝐴 = (𝐵 ∪ {𝐶})) ∨ (𝐴 = (𝐵 ∪ {𝐷}) ∨ 𝐴 = (𝐵 ∪ {𝐶, 𝐷}))))
186, 7, 173bitri 298 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 207  wa 396  wo 841   = wceq 1528  cun 3933  wss 3935  {csn 4559  {cpr 4561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-sn 4560  df-pr 4562
This theorem is referenced by:  sspr  4760  sstp  4761
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