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Theorem ssunsn2 4720
 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 4795. (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 4701 . . . . 5 (𝐷𝐴 → {𝐷} ⊆ 𝐴)
2 unss 4111 . . . . . . 7 ((𝐵𝐴 ∧ {𝐷} ⊆ 𝐴) ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴)
32bicomi 227 . . . . . 6 ((𝐵 ∪ {𝐷}) ⊆ 𝐴 ↔ (𝐵𝐴 ∧ {𝐷} ⊆ 𝐴))
43rbaibr 541 . . . . 5 ({𝐷} ⊆ 𝐴 → (𝐵𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
51, 4syl 17 . . . 4 (𝐷𝐴 → (𝐵𝐴 ↔ (𝐵 ∪ {𝐷}) ⊆ 𝐴))
65anbi1d 632 . . 3 (𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
72biimpi 219 . . . . . . 7 ((𝐵𝐴 ∧ {𝐷} ⊆ 𝐴) → (𝐵 ∪ {𝐷}) ⊆ 𝐴)
87expcom 417 . . . . . 6 ({𝐷} ⊆ 𝐴 → (𝐵𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
91, 8syl 17 . . . . 5 (𝐷𝐴 → (𝐵𝐴 → (𝐵 ∪ {𝐷}) ⊆ 𝐴))
10 ssun3 4101 . . . . . 6 (𝐴𝐶𝐴 ⊆ (𝐶 ∪ {𝐷}))
1110a1i 11 . . . . 5 (𝐷𝐴 → (𝐴𝐶𝐴 ⊆ (𝐶 ∪ {𝐷})))
129, 11anim12d 611 . . . 4 (𝐷𝐴 → ((𝐵𝐴𝐴𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
13 pm4.72 947 . . . 4 (((𝐵𝐴𝐴𝐶) → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
1412, 13sylib 221 . . 3 (𝐷𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
156, 14bitrd 282 . 2 (𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
16 uncom 4080 . . . . . . 7 ({𝐷} ∪ 𝐶) = (𝐶 ∪ {𝐷})
1716sseq2i 3944 . . . . . 6 (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ 𝐴 ⊆ (𝐶 ∪ {𝐷}))
18 ssundif 4391 . . . . . 6 (𝐴 ⊆ ({𝐷} ∪ 𝐶) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
1917, 18bitr3i 280 . . . . 5 (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶)
20 disjsn 4607 . . . . . . 7 ((𝐴 ∩ {𝐷}) = ∅ ↔ ¬ 𝐷𝐴)
21 disj3 4361 . . . . . . 7 ((𝐴 ∩ {𝐷}) = ∅ ↔ 𝐴 = (𝐴 ∖ {𝐷}))
2220, 21bitr3i 280 . . . . . 6 𝐷𝐴𝐴 = (𝐴 ∖ {𝐷}))
23 sseq1 3940 . . . . . 6 (𝐴 = (𝐴 ∖ {𝐷}) → (𝐴𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
2422, 23sylbi 220 . . . . 5 𝐷𝐴 → (𝐴𝐶 ↔ (𝐴 ∖ {𝐷}) ⊆ 𝐶))
2519, 24bitr4id 293 . . . 4 𝐷𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) ↔ 𝐴𝐶))
2625anbi2d 631 . . 3 𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ (𝐵𝐴𝐴𝐶)))
273simplbi 501 . . . . . . 7 ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐵𝐴)
2827a1i 11 . . . . . 6 𝐷𝐴 → ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐵𝐴))
2925biimpd 232 . . . . . 6 𝐷𝐴 → (𝐴 ⊆ (𝐶 ∪ {𝐷}) → 𝐴𝐶))
3028, 29anim12d 611 . . . . 5 𝐷𝐴 → (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵𝐴𝐴𝐶)))
31 pm4.72 947 . . . . 5 ((((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) → (𝐵𝐴𝐴𝐶)) ↔ ((𝐵𝐴𝐴𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶))))
3230, 31sylib 221 . . . 4 𝐷𝐴 → ((𝐵𝐴𝐴𝐶) ↔ (((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶))))
33 orcom 867 . . . 4 ((((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ∨ (𝐵𝐴𝐴𝐶)) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
3432, 33syl6bb 290 . . 3 𝐷𝐴 → ((𝐵𝐴𝐴𝐶) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
3526, 34bitrd 282 . 2 𝐷𝐴 → ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})))))
3615, 35pm2.61i 185 1 ((𝐵𝐴𝐴 ⊆ (𝐶 ∪ {𝐷})) ↔ ((𝐵𝐴𝐴𝐶) ∨ ((𝐵 ∪ {𝐷}) ⊆ 𝐴𝐴 ⊆ (𝐶 ∪ {𝐷}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  {csn 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526 This theorem is referenced by:  ssunsn  4721  ssunpr  4725  sstp  4727
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