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Theorem djuunxp 9080
Description: The union of a disjoint union and its inversion is the Cartesian product of an unordered pair and the union of the left and right classes of the disjoint unions. (Proposed by GL, 4-Jul-2022.) (Contributed by AV, 4-Jul-2022.)
Assertion
Ref Expression
djuunxp ((𝐴𝐵) ∪ (𝐵𝐴)) = ({∅, 1o} × (𝐴𝐵))

Proof of Theorem djuunxp
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djuss 9079 . . 3 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 djuss 9079 . . . 4 (𝐵𝐴) ⊆ ({∅, 1o} × (𝐵𝐴))
3 uncom 3980 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
43xpeq2i 5382 . . . 4 ({∅, 1o} × (𝐴𝐵)) = ({∅, 1o} × (𝐵𝐴))
52, 4sseqtr4i 3857 . . 3 (𝐵𝐴) ⊆ ({∅, 1o} × (𝐴𝐵))
61, 5unssi 4011 . 2 ((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ ({∅, 1o} × (𝐴𝐵))
7 elxpi 5377 . . . 4 (𝑥 ∈ ({∅, 1o} × (𝐴𝐵)) → ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))))
8 vex 3401 . . . . . . . . . 10 𝑦 ∈ V
98elpr 4421 . . . . . . . . 9 (𝑦 ∈ {∅, 1o} ↔ (𝑦 = ∅ ∨ 𝑦 = 1o))
10 elun 3976 . . . . . . . . 9 (𝑧 ∈ (𝐴𝐵) ↔ (𝑧𝐴𝑧𝐵))
11 velsn 4414 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
1211biimpri 220 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → 𝑦 ∈ {∅})
1312anim1i 608 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = ∅ ∧ 𝑧𝐴) → (𝑦 ∈ {∅} ∧ 𝑧𝐴))
1413ancoms 452 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧𝐴))
15 opelxp 5391 . . . . . . . . . . . . . . . . . 18 (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ↔ (𝑦 ∈ {∅} ∧ 𝑧𝐴))
1614, 15sylibr 226 . . . . . . . . . . . . . . . . 17 ((𝑧𝐴𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴))
1716orcd 862 . . . . . . . . . . . . . . . 16 ((𝑧𝐴𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
18 elun 3976 . . . . . . . . . . . . . . . 16 (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
1917, 18sylibr 226 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2019orcd 862 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
2120ex 403 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
2212anim1i 608 . . . . . . . . . . . . . . . . . 18 ((𝑦 = ∅ ∧ 𝑧𝐵) → (𝑦 ∈ {∅} ∧ 𝑧𝐵))
2322ancoms 452 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧𝐵))
24 opelxp 5391 . . . . . . . . . . . . . . . . 17 (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ↔ (𝑦 ∈ {∅} ∧ 𝑧𝐵))
2523, 24sylibr 226 . . . . . . . . . . . . . . . 16 ((𝑧𝐵𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵))
2625orcd 862 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
2726olcd 863 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
2827ex 403 . . . . . . . . . . . . 13 (𝑧𝐵 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
2921, 28jaoi 846 . . . . . . . . . . . 12 ((𝑧𝐴𝑧𝐵) → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
3029com12 32 . . . . . . . . . . 11 (𝑦 = ∅ → ((𝑧𝐴𝑧𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
31 velsn 4414 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {1o} ↔ 𝑦 = 1o)
3231biimpri 220 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 1o𝑦 ∈ {1o})
3332anim1i 608 . . . . . . . . . . . . . . . . . 18 ((𝑦 = 1o𝑧𝐴) → (𝑦 ∈ {1o} ∧ 𝑧𝐴))
3433ancoms 452 . . . . . . . . . . . . . . . . 17 ((𝑧𝐴𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧𝐴))
35 opelxp 5391 . . . . . . . . . . . . . . . . 17 (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴) ↔ (𝑦 ∈ {1o} ∧ 𝑧𝐴))
3634, 35sylibr 226 . . . . . . . . . . . . . . . 16 ((𝑧𝐴𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))
3736olcd 863 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
3837olcd 863 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
3938ex 403 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
4032anim1i 608 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = 1o𝑧𝐵) → (𝑦 ∈ {1o} ∧ 𝑧𝐵))
4140ancoms 452 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐵𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧𝐵))
42 opelxp 5391 . . . . . . . . . . . . . . . . . 18 (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵) ↔ (𝑦 ∈ {1o} ∧ 𝑧𝐵))
4341, 42sylibr 226 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵))
4443olcd 863 . . . . . . . . . . . . . . . 16 ((𝑧𝐵𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
4544, 18sylibr 226 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
4645orcd 862 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
4746ex 403 . . . . . . . . . . . . 13 (𝑧𝐵 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
4839, 47jaoi 846 . . . . . . . . . . . 12 ((𝑧𝐴𝑧𝐵) → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
4948com12 32 . . . . . . . . . . 11 (𝑦 = 1o → ((𝑧𝐴𝑧𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
5030, 49jaoi 846 . . . . . . . . . 10 ((𝑦 = ∅ ∨ 𝑦 = 1o) → ((𝑧𝐴𝑧𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
5150imp 397 . . . . . . . . 9 (((𝑦 = ∅ ∨ 𝑦 = 1o) ∧ (𝑧𝐴𝑧𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
529, 10, 51syl2anb 591 . . . . . . . 8 ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
53 elun 3976 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴)))
54 df-dju 9061 . . . . . . . . . . 11 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
5554eleq2i 2851 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
56 df-dju 9061 . . . . . . . . . . . 12 (𝐵𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴))
5756eleq2i 2851 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)))
58 elun 3976 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
5957, 58bitri 267 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
6055, 59orbi12i 901 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
6153, 60bitri 267 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
6252, 61sylibr 226 . . . . . . 7 ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵)) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
6362adantl 475 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
64 eleq1 2847 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴))))
6564adantr 474 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴))))
6663, 65mpbird 249 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → 𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
6766exlimivv 1975 . . . 4 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → 𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
687, 67syl 17 . . 3 (𝑥 ∈ ({∅, 1o} × (𝐴𝐵)) → 𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
6968ssriv 3825 . 2 ({∅, 1o} × (𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ (𝐵𝐴))
706, 69eqssi 3837 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 836   = wceq 1601  wex 1823  wcel 2107  cun 3790  c0 4141  {csn 4398  {cpr 4400  cop 4404   × cxp 5353  1oc1o 7836  cdju 9058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fv 6143  df-1st 7445  df-2nd 7446  df-1o 7843  df-dju 9061  df-inl 9062  df-inr 9063
This theorem is referenced by: (None)
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