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Theorem ssunsn2 3865
 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 3884. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
ssunsn2 ((B A A (C ∪ {D})) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D}))))

Proof of Theorem ssunsn2
StepHypRef Expression
1 snssi 3852 . . . . 5 (D A → {D} A)
2 unss 3437 . . . . . . 7 ((B A {D} A) ↔ (B ∪ {D}) A)
32bicomi 193 . . . . . 6 ((B ∪ {D}) A ↔ (B A {D} A))
43rbaibr 874 . . . . 5 ({D} A → (B A ↔ (B ∪ {D}) A))
51, 4syl 15 . . . 4 (D A → (B A ↔ (B ∪ {D}) A))
65anbi1d 685 . . 3 (D A → ((B A A (C ∪ {D})) ↔ ((B ∪ {D}) A A (C ∪ {D}))))
72biimpi 186 . . . . . . 7 ((B A {D} A) → (B ∪ {D}) A)
87expcom 424 . . . . . 6 ({D} A → (B A → (B ∪ {D}) A))
91, 8syl 15 . . . . 5 (D A → (B A → (B ∪ {D}) A))
10 ssun3 3428 . . . . . 6 (A CA (C ∪ {D}))
1110a1i 10 . . . . 5 (D A → (A CA (C ∪ {D})))
129, 11anim12d 546 . . . 4 (D A → ((B A A C) → ((B ∪ {D}) A A (C ∪ {D}))))
13 pm4.72 846 . . . 4 (((B A A C) → ((B ∪ {D}) A A (C ∪ {D}))) ↔ (((B ∪ {D}) A A (C ∪ {D})) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D})))))
1412, 13sylib 188 . . 3 (D A → (((B ∪ {D}) A A (C ∪ {D})) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D})))))
156, 14bitrd 244 . 2 (D A → ((B A A (C ∪ {D})) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D})))))
16 disjsn 3786 . . . . . . 7 ((A ∩ {D}) = ↔ ¬ D A)
17 disj3 3595 . . . . . . 7 ((A ∩ {D}) = A = (A {D}))
1816, 17bitr3i 242 . . . . . 6 D AA = (A {D}))
19 sseq1 3292 . . . . . 6 (A = (A {D}) → (A C ↔ (A {D}) C))
2018, 19sylbi 187 . . . . 5 D A → (A C ↔ (A {D}) C))
21 uncom 3408 . . . . . . 7 ({D} ∪ C) = (C ∪ {D})
2221sseq2i 3296 . . . . . 6 (A ({D} ∪ C) ↔ A (C ∪ {D}))
23 ssundif 3633 . . . . . 6 (A ({D} ∪ C) ↔ (A {D}) C)
2422, 23bitr3i 242 . . . . 5 (A (C ∪ {D}) ↔ (A {D}) C)
2520, 24syl6rbbr 255 . . . 4 D A → (A (C ∪ {D}) ↔ A C))
2625anbi2d 684 . . 3 D A → ((B A A (C ∪ {D})) ↔ (B A A C)))
273simplbi 446 . . . . . . 7 ((B ∪ {D}) AB A)
2827a1i 10 . . . . . 6 D A → ((B ∪ {D}) AB A))
2925biimpd 198 . . . . . 6 D A → (A (C ∪ {D}) → A C))
3028, 29anim12d 546 . . . . 5 D A → (((B ∪ {D}) A A (C ∪ {D})) → (B A A C)))
31 pm4.72 846 . . . . 5 ((((B ∪ {D}) A A (C ∪ {D})) → (B A A C)) ↔ ((B A A C) ↔ (((B ∪ {D}) A A (C ∪ {D})) (B A A C))))
3230, 31sylib 188 . . . 4 D A → ((B A A C) ↔ (((B ∪ {D}) A A (C ∪ {D})) (B A A C))))
33 orcom 376 . . . 4 ((((B ∪ {D}) A A (C ∪ {D})) (B A A C)) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D}))))
3432, 33syl6bb 252 . . 3 D A → ((B A A C) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D})))))
3526, 34bitrd 244 . 2 D A → ((B A A (C ∪ {D})) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D})))))
3615, 35pm2.61i 156 1 ((B A A (C ∪ {D})) ↔ ((B A A C) ((B ∪ {D}) A A (C ∪ {D}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3206   ∪ cun 3207   ∩ cin 3208   ⊆ wss 3257  ∅c0 3550  {csn 3737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741 This theorem is referenced by:  ssunsn  3866  ssunpr  3868  sstp  3870
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