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Theorem ssunsn2 4494
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 4570. (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 4475 . . . . 5 (𝐷𝐴 → {𝐷} ⊆ 𝐴)
2 unss 3939 . . . . . . 7 ((𝐵𝐴 ∧ {𝐷} ⊆ 𝐴) ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴)
32bicomi 214 . . . . . 6 ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ↔ (𝐵𝐴 ∧ {𝐷} ⊆ 𝐴))
43rbaibr 521 . . . . 5 ({𝐷} ⊆ 𝐴 → (𝐵𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
51, 4syl 17 . . . 4 (𝐷𝐴 → (𝐵𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
65anbi1d 609 . . 3 (𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
72biimpi 206 . . . . . . 7 ((𝐵𝐴 ∧ {𝐷} ⊆ 𝐴) → (𝐵 ∪ {𝐷}) ⊆ 𝐴)
87expcom 398 . . . . . 6 ({𝐷} ⊆ 𝐴 → (𝐵𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
91, 8syl 17 . . . . 5 (𝐷𝐴 → (𝐵𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
10 ssun3 3930 . . . . . 6 (𝐴𝐶𝐴 ⊆ (𝐶 ∪ {𝐷}))
1110a1i 11 . . . . 5 (𝐷𝐴 → (𝐴𝐶𝐴 ⊆ (𝐶 ∪ {𝐷})))
129, 11anim12d 590 . . . 4 (𝐷𝐴 → ((𝐵𝐴𝐴𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
13 pm4.72 926 . . . 4 (((𝐵𝐴𝐴𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
1412, 13sylib 208 . . 3 (𝐷𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
156, 14bitrd 268 . 2 (𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
16 disjsn 4384 . . . . . . 7 ((𝐴 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷𝐴)
17 disj3 4165 . . . . . . 7 ((𝐴 ∩ {𝐷}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐷}))
1816, 17bitr3i 266 . . . . . 6 𝐷𝐴𝐴 = (𝐴 ∖ {𝐷}))
19 sseq1 3776 . . . . . 6 (𝐴 = (𝐴 ∖ {𝐷}) → (𝐴𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
2018, 19sylbi 207 . . . . 5 𝐷𝐴 → (𝐴𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
21 uncom 3909 . . . . . . 7 ({𝐷} ∪ 𝐶) = (𝐶 ∪ {𝐷})
2221sseq2i 3780 . . . . . 6 (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ {𝐷}))
23 ssundif 4195 . . . . . 6 (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
2422, 23bitr3i 266 . . . . 5 (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
2520, 24syl6rbbr 279 . . . 4 𝐷𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ 𝐴𝐶))
2625anbi2d 608 . . 3 𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ (𝐵𝐴𝐴𝐶)))
273simplbi 481 . . . . . . 7 ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐵𝐴)
2827a1i 11 . . . . . 6 𝐷𝐴 → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐵𝐴))
2925biimpd 219 . . . . . 6 𝐷𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) → 𝐴𝐶))
3028, 29anim12d 590 . . . . 5 𝐷𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵𝐴𝐴𝐶)))
31 pm4.72 926 . . . . 5 ((((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵𝐴𝐴𝐶)) ↔ ((𝐵𝐴𝐴𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶))))
3230, 31sylib 208 . . . 4 𝐷𝐴 → ((𝐵𝐴𝐴𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶))))
33 orcom 851 . . . 4 ((((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶)) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
3432, 33syl6bb 276 . . 3 𝐷𝐴 → ((𝐵𝐴𝐴𝐶) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
3526, 34bitrd 268 . 2 𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
3615, 35pm2.61i 176 1 ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 828   = wceq 1631  wcel 2145  cdif 3721  cun 3722  cin 3723  wss 3724  c0 4064  {csn 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-sn 4318
This theorem is referenced by:  ssunsn  4495  ssunpr  4499  sstp  4501
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