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Theorem djuss 9960
Description: A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
Assertion
Ref Expression
djuss (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))

Proof of Theorem djuss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djur 9959 . . 3 (𝑥 ∈ (𝐴𝐵) → (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
2 simpr 484 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3 df-inl 9942 . . . . . . . . 9 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4 opeq2 4874 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5 elex 3501 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ V)
6 opex 5469 . . . . . . . . . 10 ⟨∅, 𝑦⟩ ∈ V
76a1i 11 . . . . . . . . 9 (𝑦𝐴 → ⟨∅, 𝑦⟩ ∈ V)
83, 4, 5, 7fvmptd3 7039 . . . . . . . 8 (𝑦𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
98adantr 480 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
102, 9eqtrd 2777 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
11 elun1 4182 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ (𝐴𝐵))
12 0ex 5307 . . . . . . . . . 10 ∅ ∈ V
1312prid1 4762 . . . . . . . . 9 ∅ ∈ {∅, 1o}
1411, 13jctil 519 . . . . . . . 8 (𝑦𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1514adantr 480 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
16 opelxp 5721 . . . . . . 7 (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1715, 16sylibr 234 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
1810, 17eqeltrd 2841 . . . . 5 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
1918rexlimiva 3147 . . . 4 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
20 simpr 484 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
21 df-inr 9943 . . . . . . . . 9 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
22 opeq2 4874 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
23 elex 3501 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ V)
24 opex 5469 . . . . . . . . . 10 ⟨1o, 𝑦⟩ ∈ V
2524a1i 11 . . . . . . . . 9 (𝑦𝐵 → ⟨1o, 𝑦⟩ ∈ V)
2621, 22, 23, 25fvmptd3 7039 . . . . . . . 8 (𝑦𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2726adantr 480 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2820, 27eqtrd 2777 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
29 elun2 4183 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ (𝐴𝐵))
3029adantr 480 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴𝐵))
31 1oex 8516 . . . . . . . . 9 1o ∈ V
3231prid2 4763 . . . . . . . 8 1o ∈ {∅, 1o}
3330, 32jctil 519 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
34 opelxp 5721 . . . . . . 7 (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
3533, 34sylibr 234 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
3628, 35eqeltrd 2841 . . . . 5 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3736rexlimiva 3147 . . . 4 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3819, 37jaoi 858 . . 3 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
391, 38syl 17 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
4039ssriv 3987 1 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 848   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cun 3949  wss 3951  c0 4333  {cpr 4628  cop 4632   × cxp 5683  cfv 6561  1oc1o 8499  cdju 9938  inlcinl 9939  inrcinr 9940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-dju 9941  df-inl 9942  df-inr 9943
This theorem is referenced by:  djuunxp  9961  djuexALT  9962  eldju1st  9963
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