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Theorem djuss 6668
Description: A disjoint union is a subset 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 6664 . . 3 (𝑥 ∈ (𝐴𝐵) → (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
2 simpr 108 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3 df-inl 6646 . . . . . . . . . 10 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
43a1i 9 . . . . . . . . 9 (𝑦𝐴 → inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩))
5 opeq2 3597 . . . . . . . . . 10 (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
65adantl 271 . . . . . . . . 9 ((𝑦𝐴𝑥 = 𝑦) → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
7 elex 2621 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ V)
8 0ex 3931 . . . . . . . . . . 11 ∅ ∈ V
9 vex 2615 . . . . . . . . . . 11 𝑦 ∈ V
108, 9opex 4020 . . . . . . . . . 10 ⟨∅, 𝑦⟩ ∈ V
1110a1i 9 . . . . . . . . 9 (𝑦𝐴 → ⟨∅, 𝑦⟩ ∈ V)
124, 6, 7, 11fvmptd 5330 . . . . . . . 8 (𝑦𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
1312adantr 270 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
142, 13eqtrd 2115 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
15 elun1 3151 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ (𝐴𝐵))
168prid1 3522 . . . . . . . . 9 ∅ ∈ {∅, 1𝑜}
1715, 16jctil 305 . . . . . . . 8 (𝑦𝐴 → (∅ ∈ {∅, 1𝑜} ∧ 𝑦 ∈ (𝐴𝐵)))
1817adantr 270 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1𝑜} ∧ 𝑦 ∈ (𝐴𝐵)))
19 opelxp 4430 . . . . . . 7 (⟨∅, 𝑦⟩ ∈ ({∅, 1𝑜} × (𝐴𝐵)) ↔ (∅ ∈ {∅, 1𝑜} ∧ 𝑦 ∈ (𝐴𝐵)))
2018, 19sylibr 132 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1𝑜} × (𝐴𝐵)))
2114, 20eqeltrd 2159 . . . . 5 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1𝑜} × (𝐴𝐵)))
2221rexlimiva 2478 . . . 4 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1𝑜} × (𝐴𝐵)))
23 simpr 108 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
24 df-inr 6647 . . . . . . . . . 10 inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩)
2524a1i 9 . . . . . . . . 9 (𝑦𝐵 → inr = (𝑥 ∈ V ↦ ⟨1𝑜, 𝑥⟩))
26 opeq2 3597 . . . . . . . . . 10 (𝑥 = 𝑦 → ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, 𝑦⟩)
2726adantl 271 . . . . . . . . 9 ((𝑦𝐵𝑥 = 𝑦) → ⟨1𝑜, 𝑥⟩ = ⟨1𝑜, 𝑦⟩)
28 elex 2621 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ V)
29 1oex 6121 . . . . . . . . . . 11 1𝑜 ∈ V
3029, 9opex 4020 . . . . . . . . . 10 ⟨1𝑜, 𝑦⟩ ∈ V
3130a1i 9 . . . . . . . . 9 (𝑦𝐵 → ⟨1𝑜, 𝑦⟩ ∈ V)
3225, 27, 28, 31fvmptd 5330 . . . . . . . 8 (𝑦𝐵 → (inr‘𝑦) = ⟨1𝑜, 𝑦⟩)
3332adantr 270 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1𝑜, 𝑦⟩)
3423, 33eqtrd 2115 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1𝑜, 𝑦⟩)
35 elun2 3152 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ (𝐴𝐵))
3635adantr 270 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴𝐵))
3729prid2 3523 . . . . . . . 8 1𝑜 ∈ {∅, 1𝑜}
3836, 37jctil 305 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (1𝑜 ∈ {∅, 1𝑜} ∧ 𝑦 ∈ (𝐴𝐵)))
39 opelxp 4430 . . . . . . 7 (⟨1𝑜, 𝑦⟩ ∈ ({∅, 1𝑜} × (𝐴𝐵)) ↔ (1𝑜 ∈ {∅, 1𝑜} ∧ 𝑦 ∈ (𝐴𝐵)))
4038, 39sylibr 132 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → ⟨1𝑜, 𝑦⟩ ∈ ({∅, 1𝑜} × (𝐴𝐵)))
4134, 40eqeltrd 2159 . . . . 5 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1𝑜} × (𝐴𝐵)))
4241rexlimiva 2478 . . . 4 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1𝑜} × (𝐴𝐵)))
4322, 42jaoi 669 . . 3 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1𝑜} × (𝐴𝐵)))
441, 43syl 14 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ({∅, 1𝑜} × (𝐴𝐵)))
4544ssriv 3014 1 (𝐴𝐵) ⊆ ({∅, 1𝑜} × (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wcel 1434  wrex 2354  Vcvv 2612  cun 2982  wss 2984  c0 3269  {cpr 3423  cop 3425  cmpt 3865   × cxp 4399  cfv 4969  1𝑜c1o 6106  cdju 6637  inlcinl 6644  inrcinr 6645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fo 4975  df-fv 4977  df-1st 5846  df-2nd 5847  df-1o 6113  df-dju 6638  df-inl 6646  df-inr 6647
This theorem is referenced by:  eldju1st  6669
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