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Theorem djuss 9856
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 9855 . . 3 (𝑥 ∈ (𝐴𝐵) → (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
2 simpr 485 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3 df-inl 9838 . . . . . . . . 9 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4 opeq2 4831 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5 elex 3463 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ V)
6 opex 5421 . . . . . . . . . 10 ⟨∅, 𝑦⟩ ∈ V
76a1i 11 . . . . . . . . 9 (𝑦𝐴 → ⟨∅, 𝑦⟩ ∈ V)
83, 4, 5, 7fvmptd3 6971 . . . . . . . 8 (𝑦𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
98adantr 481 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
102, 9eqtrd 2776 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
11 elun1 4136 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ (𝐴𝐵))
12 0ex 5264 . . . . . . . . . 10 ∅ ∈ V
1312prid1 4723 . . . . . . . . 9 ∅ ∈ {∅, 1o}
1411, 13jctil 520 . . . . . . . 8 (𝑦𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1514adantr 481 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
16 opelxp 5669 . . . . . . 7 (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1715, 16sylibr 233 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
1810, 17eqeltrd 2838 . . . . 5 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
1918rexlimiva 3144 . . . 4 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
20 simpr 485 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
21 df-inr 9839 . . . . . . . . 9 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
22 opeq2 4831 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
23 elex 3463 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ V)
24 opex 5421 . . . . . . . . . 10 ⟨1o, 𝑦⟩ ∈ V
2524a1i 11 . . . . . . . . 9 (𝑦𝐵 → ⟨1o, 𝑦⟩ ∈ V)
2621, 22, 23, 25fvmptd3 6971 . . . . . . . 8 (𝑦𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2726adantr 481 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2820, 27eqtrd 2776 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
29 elun2 4137 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ (𝐴𝐵))
3029adantr 481 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴𝐵))
31 1oex 8422 . . . . . . . . 9 1o ∈ V
3231prid2 4724 . . . . . . . 8 1o ∈ {∅, 1o}
3330, 32jctil 520 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
34 opelxp 5669 . . . . . . 7 (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
3533, 34sylibr 233 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
3628, 35eqeltrd 2838 . . . . 5 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3736rexlimiva 3144 . . . 4 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3819, 37jaoi 855 . . 3 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
391, 38syl 17 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
4039ssriv 3948 1 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 845   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cun 3908  wss 3910  c0 4282  {cpr 4588  cop 4592   × cxp 5631  cfv 6496  1oc1o 8405  cdju 9834  inlcinl 9835  inrcinr 9836
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-suc 6323  df-iota 6448  df-fun 6498  df-fv 6504  df-1st 7921  df-2nd 7922  df-1o 8412  df-dju 9837  df-inl 9838  df-inr 9839
This theorem is referenced by:  djuunxp  9857  djuexALT  9858  eldju1st  9859
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