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Theorem djuunxp 9336
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 9335 . . 3 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 djuss 9335 . . . 4 (𝐵𝐴) ⊆ ({∅, 1o} × (𝐵𝐴))
3 uncom 4113 . . . . 5 (𝐴𝐵) = (𝐵𝐴)
43xpeq2i 5565 . . . 4 ({∅, 1o} × (𝐴𝐵)) = ({∅, 1o} × (𝐵𝐴))
52, 4sseqtrri 3988 . . 3 (𝐵𝐴) ⊆ ({∅, 1o} × (𝐴𝐵))
61, 5unssi 4145 . 2 ((𝐴𝐵) ∪ (𝐵𝐴)) ⊆ ({∅, 1o} × (𝐴𝐵))
7 elxpi 5560 . . . 4 (𝑥 ∈ ({∅, 1o} × (𝐴𝐵)) → ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))))
8 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
98elpr 4571 . . . . . . . . 9 (𝑦 ∈ {∅, 1o} ↔ (𝑦 = ∅ ∨ 𝑦 = 1o))
10 elun 4109 . . . . . . . . 9 (𝑧 ∈ (𝐴𝐵) ↔ (𝑧𝐴𝑧𝐵))
11 velsn 4564 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
1211biimpri 231 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ∅ → 𝑦 ∈ {∅})
1312anim1i 617 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = ∅ ∧ 𝑧𝐴) → (𝑦 ∈ {∅} ∧ 𝑧𝐴))
1413ancoms 462 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐴𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧𝐴))
15 opelxp 5574 . . . . . . . . . . . . . . . . . 18 (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ↔ (𝑦 ∈ {∅} ∧ 𝑧𝐴))
1614, 15sylibr 237 . . . . . . . . . . . . . . . . 17 ((𝑧𝐴𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴))
1716orcd 870 . . . . . . . . . . . . . . . 16 ((𝑧𝐴𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
18 elun 4109 . . . . . . . . . . . . . . . 16 (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
1917, 18sylibr 237 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2019orcd 870 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
2120ex 416 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
2212anim1i 617 . . . . . . . . . . . . . . . . . 18 ((𝑦 = ∅ ∧ 𝑧𝐵) → (𝑦 ∈ {∅} ∧ 𝑧𝐵))
2322ancoms 462 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑦 = ∅) → (𝑦 ∈ {∅} ∧ 𝑧𝐵))
24 opelxp 5574 . . . . . . . . . . . . . . . . 17 (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ↔ (𝑦 ∈ {∅} ∧ 𝑧𝐵))
2523, 24sylibr 237 . . . . . . . . . . . . . . . 16 ((𝑧𝐵𝑦 = ∅) → ⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵))
2625orcd 870 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
2726olcd 871 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑦 = ∅) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
2827ex 416 . . . . . . . . . . . . 13 (𝑧𝐵 → (𝑦 = ∅ → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
2921, 28jaoi 854 . . . . . . . . . . . 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 4564 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ {1o} ↔ 𝑦 = 1o)
3231biimpri 231 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 1o𝑦 ∈ {1o})
3332anim1i 617 . . . . . . . . . . . . . . . . . 18 ((𝑦 = 1o𝑧𝐴) → (𝑦 ∈ {1o} ∧ 𝑧𝐴))
3433ancoms 462 . . . . . . . . . . . . . . . . 17 ((𝑧𝐴𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧𝐴))
35 opelxp 5574 . . . . . . . . . . . . . . . . 17 (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴) ↔ (𝑦 ∈ {1o} ∧ 𝑧𝐴))
3634, 35sylibr 237 . . . . . . . . . . . . . . . 16 ((𝑧𝐴𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))
3736olcd 871 . . . . . . . . . . . . . . 15 ((𝑧𝐴𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
3837olcd 871 . . . . . . . . . . . . . 14 ((𝑧𝐴𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
3938ex 416 . . . . . . . . . . . . 13 (𝑧𝐴 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
4032anim1i 617 . . . . . . . . . . . . . . . . . . 19 ((𝑦 = 1o𝑧𝐵) → (𝑦 ∈ {1o} ∧ 𝑧𝐵))
4140ancoms 462 . . . . . . . . . . . . . . . . . 18 ((𝑧𝐵𝑦 = 1o) → (𝑦 ∈ {1o} ∧ 𝑧𝐵))
42 opelxp 5574 . . . . . . . . . . . . . . . . . 18 (⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵) ↔ (𝑦 ∈ {1o} ∧ 𝑧𝐵))
4341, 42sylibr 237 . . . . . . . . . . . . . . . . 17 ((𝑧𝐵𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵))
4443olcd 871 . . . . . . . . . . . . . . . 16 ((𝑧𝐵𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐴) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐵)))
4544, 18sylibr 237 . . . . . . . . . . . . . . 15 ((𝑧𝐵𝑦 = 1o) → ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
4645orcd 870 . . . . . . . . . . . . . 14 ((𝑧𝐵𝑦 = 1o) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
4746ex 416 . . . . . . . . . . . . 13 (𝑧𝐵 → (𝑦 = 1o → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
4839, 47jaoi 854 . . . . . . . . . . . 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 854 . . . . . . . . . 10 ((𝑦 = ∅ ∨ 𝑦 = 1o) → ((𝑧𝐴𝑧𝐵) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))))
5150imp 410 . . . . . . . . 9 (((𝑦 = ∅ ∨ 𝑦 = 1o) ∧ (𝑧𝐴𝑧𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
529, 10, 51syl2anb 600 . . . . . . . 8 ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵)) → (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
53 elun 4109 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴)))
54 df-dju 9316 . . . . . . . . . . 11 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
5554eleq2i 2907 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
56 df-dju 9316 . . . . . . . . . . . 12 (𝐵𝐴) = (({∅} × 𝐵) ∪ ({1o} × 𝐴))
5756eleq2i 2907 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)))
58 elun 4109 . . . . . . . . . . 11 (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐵) ∪ ({1o} × 𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
5957, 58bitri 278 . . . . . . . . . 10 (⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴) ↔ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴)))
6055, 59orbi12i 912 . . . . . . . . 9 ((⟨𝑦, 𝑧⟩ ∈ (𝐴𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ (𝐵𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
6153, 60bitri 278 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ (⟨𝑦, 𝑧⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ∨ (⟨𝑦, 𝑧⟩ ∈ ({∅} × 𝐵) ∨ ⟨𝑦, 𝑧⟩ ∈ ({1o} × 𝐴))))
6252, 61sylibr 237 . . . . . . 7 ((𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵)) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
6362adantl 485 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
64 eleq1 2903 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴))))
6564adantr 484 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → (𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)) ↔ ⟨𝑦, 𝑧⟩ ∈ ((𝐴𝐵) ∪ (𝐵𝐴))))
6663, 65mpbird 260 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → 𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
6766exlimivv 1934 . . . 4 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ {∅, 1o} ∧ 𝑧 ∈ (𝐴𝐵))) → 𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
687, 67syl 17 . . 3 (𝑥 ∈ ({∅, 1o} × (𝐴𝐵)) → 𝑥 ∈ ((𝐴𝐵) ∪ (𝐵𝐴)))
6968ssriv 3955 . 2 ({∅, 1o} × (𝐴𝐵)) ⊆ ((𝐴𝐵) ∪ (𝐵𝐴))
706, 69eqssi 3967 1 ((𝐴𝐵) ∪ (𝐵𝐴)) = ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2115  cun 3916  c0 4274  {csn 4548  {cpr 4550  cop 4554   × cxp 5536  1oc1o 8080  cdju 9313
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6297  df-fun 6340  df-fv 6346  df-1st 7674  df-2nd 7675  df-1o 8087  df-dju 9316  df-inl 9317  df-inr 9318
This theorem is referenced by: (None)
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