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Theorem djuss 7009
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 7008 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
2 simpr 109 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3 df-inl 6986 . . . . . . . . 9 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4 opeq2 3742 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5 elex 2723 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ V)
6 0ex 4091 . . . . . . . . . . 11 ∅ ∈ V
7 vex 2715 . . . . . . . . . . 11 𝑦 ∈ V
86, 7opex 4189 . . . . . . . . . 10 ⟨∅, 𝑦⟩ ∈ V
98a1i 9 . . . . . . . . 9 (𝑦𝐴 → ⟨∅, 𝑦⟩ ∈ V)
103, 4, 5, 9fvmptd3 5560 . . . . . . . 8 (𝑦𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
1110adantr 274 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
122, 11eqtrd 2190 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
13 elun1 3274 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ (𝐴𝐵))
146prid1 3665 . . . . . . . . 9 ∅ ∈ {∅, 1o}
1513, 14jctil 310 . . . . . . . 8 (𝑦𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1615adantr 274 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
17 opelxp 4615 . . . . . . 7 (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1816, 17sylibr 133 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
1912, 18eqeltrd 2234 . . . . 5 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
2019rexlimiva 2569 . . . 4 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
21 simpr 109 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
22 df-inr 6987 . . . . . . . . 9 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
23 opeq2 3742 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
24 elex 2723 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ V)
25 1oex 6368 . . . . . . . . . . 11 1o ∈ V
2625, 7opex 4189 . . . . . . . . . 10 ⟨1o, 𝑦⟩ ∈ V
2726a1i 9 . . . . . . . . 9 (𝑦𝐵 → ⟨1o, 𝑦⟩ ∈ V)
2822, 23, 24, 27fvmptd3 5560 . . . . . . . 8 (𝑦𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2928adantr 274 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
3021, 29eqtrd 2190 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
31 elun2 3275 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ (𝐴𝐵))
3231adantr 274 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴𝐵))
3325prid2 3666 . . . . . . . 8 1o ∈ {∅, 1o}
3432, 33jctil 310 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
35 opelxp 4615 . . . . . . 7 (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
3634, 35sylibr 133 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
3730, 36eqeltrd 2234 . . . . 5 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3837rexlimiva 2569 . . . 4 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3920, 38jaoi 706 . . 3 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
401, 39sylbi 120 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
4140ssriv 3132 1 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wo 698   = wceq 1335  wcel 2128  wrex 2436  Vcvv 2712  cun 3100  wss 3102  c0 3394  {cpr 3561  cop 3563   × cxp 4583  cfv 5169  1oc1o 6353  cdju 6976  inlcinl 6984  inrcinr 6985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1st 6085  df-2nd 6086  df-1o 6360  df-dju 6977  df-inl 6986  df-inr 6987
This theorem is referenced by:  eldju1st  7010
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