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Theorem ssunsn2 4832
Description: The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4907. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
ssunsn2 ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))

Proof of Theorem ssunsn2
StepHypRef Expression
1 snssi 4813 . . . . 5 (𝐷𝐴 → {𝐷} ⊆ 𝐴)
2 unss 4200 . . . . . . 7 ((𝐵𝐴 ∧ {𝐷} ⊆ 𝐴) ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴)
32bicomi 224 . . . . . 6 ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ↔ (𝐵𝐴 ∧ {𝐷} ⊆ 𝐴))
43rbaibr 537 . . . . 5 ({𝐷} ⊆ 𝐴 → (𝐵𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
51, 4syl 17 . . . 4 (𝐷𝐴 → (𝐵𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
65anbi1d 631 . . 3 (𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
72biimpi 216 . . . . . . 7 ((𝐵𝐴 ∧ {𝐷} ⊆ 𝐴) → (𝐵 ∪ {𝐷}) ⊆ 𝐴)
87expcom 413 . . . . . 6 ({𝐷} ⊆ 𝐴 → (𝐵𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
91, 8syl 17 . . . . 5 (𝐷𝐴 → (𝐵𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
10 ssun3 4190 . . . . . 6 (𝐴𝐶𝐴 ⊆ (𝐶 ∪ {𝐷}))
1110a1i 11 . . . . 5 (𝐷𝐴 → (𝐴𝐶𝐴 ⊆ (𝐶 ∪ {𝐷})))
129, 11anim12d 609 . . . 4 (𝐷𝐴 → ((𝐵𝐴𝐴𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
13 pm4.72 951 . . . 4 (((𝐵𝐴𝐴𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
1412, 13sylib 218 . . 3 (𝐷𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
156, 14bitrd 279 . 2 (𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
16 uncom 4168 . . . . . . 7 ({𝐷} ∪ 𝐶) = (𝐶 ∪ {𝐷})
1716sseq2i 4025 . . . . . 6 (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ {𝐷}))
18 ssundif 4494 . . . . . 6 (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
1917, 18bitr3i 277 . . . . 5 (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
20 disjsn 4716 . . . . . . 7 ((𝐴 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷𝐴)
21 disj3 4460 . . . . . . 7 ((𝐴 ∩ {𝐷}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐷}))
2220, 21bitr3i 277 . . . . . 6 𝐷𝐴𝐴 = (𝐴 ∖ {𝐷}))
23 sseq1 4021 . . . . . 6 (𝐴 = (𝐴 ∖ {𝐷}) → (𝐴𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
2422, 23sylbi 217 . . . . 5 𝐷𝐴 → (𝐴𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
2519, 24bitr4id 290 . . . 4 𝐷𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ 𝐴𝐶))
2625anbi2d 630 . . 3 𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ (𝐵𝐴𝐴𝐶)))
273simplbi 497 . . . . . . 7 ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐵𝐴)
2827a1i 11 . . . . . 6 𝐷𝐴 → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐵𝐴))
2925biimpd 229 . . . . . 6 𝐷𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) → 𝐴𝐶))
3028, 29anim12d 609 . . . . 5 𝐷𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵𝐴𝐴𝐶)))
31 pm4.72 951 . . . . 5 ((((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵𝐴𝐴𝐶)) ↔ ((𝐵𝐴𝐴𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶))))
3230, 31sylib 218 . . . 4 𝐷𝐴 → ((𝐵𝐴𝐴𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶))))
33 orcom 870 . . . 4 ((((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶)) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
3432, 33bitrdi 287 . . 3 𝐷𝐴 → ((𝐵𝐴𝐴𝐶) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
3526, 34bitrd 279 . 2 𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
3615, 35pm2.61i 182 1 ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632
This theorem is referenced by:  ssunsn  4833  ssunpr  4839  sstp  4841
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