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Theorem djuss 6948
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 6947 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
2 simpr 109 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3 df-inl 6925 . . . . . . . . 9 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4 opeq2 3701 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5 elex 2692 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ V)
6 0ex 4050 . . . . . . . . . . 11 ∅ ∈ V
7 vex 2684 . . . . . . . . . . 11 𝑦 ∈ V
86, 7opex 4146 . . . . . . . . . 10 ⟨∅, 𝑦⟩ ∈ V
98a1i 9 . . . . . . . . 9 (𝑦𝐴 → ⟨∅, 𝑦⟩ ∈ V)
103, 4, 5, 9fvmptd3 5507 . . . . . . . 8 (𝑦𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
1110adantr 274 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
122, 11eqtrd 2170 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
13 elun1 3238 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ (𝐴𝐵))
146prid1 3624 . . . . . . . . 9 ∅ ∈ {∅, 1o}
1513, 14jctil 310 . . . . . . . 8 (𝑦𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1615adantr 274 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
17 opelxp 4564 . . . . . . 7 (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1816, 17sylibr 133 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
1912, 18eqeltrd 2214 . . . . 5 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
2019rexlimiva 2542 . . . 4 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
21 simpr 109 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
22 df-inr 6926 . . . . . . . . 9 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
23 opeq2 3701 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
24 elex 2692 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ V)
25 1oex 6314 . . . . . . . . . . 11 1o ∈ V
2625, 7opex 4146 . . . . . . . . . 10 ⟨1o, 𝑦⟩ ∈ V
2726a1i 9 . . . . . . . . 9 (𝑦𝐵 → ⟨1o, 𝑦⟩ ∈ V)
2822, 23, 24, 27fvmptd3 5507 . . . . . . . 8 (𝑦𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2928adantr 274 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
3021, 29eqtrd 2170 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
31 elun2 3239 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ (𝐴𝐵))
3231adantr 274 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴𝐵))
3325prid2 3625 . . . . . . . 8 1o ∈ {∅, 1o}
3432, 33jctil 310 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
35 opelxp 4564 . . . . . . 7 (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
3634, 35sylibr 133 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
3730, 36eqeltrd 2214 . . . . 5 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3837rexlimiva 2542 . . . 4 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3920, 38jaoi 705 . . 3 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
401, 39sylbi 120 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
4140ssriv 3096 1 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wo 697   = wceq 1331  wcel 1480  wrex 2415  Vcvv 2681  cun 3064  wss 3066  c0 3358  {cpr 3523  cop 3525   × cxp 4532  cfv 5118  1oc1o 6299  cdju 6915  inlcinl 6923  inrcinr 6924
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dju 6916  df-inl 6925  df-inr 6926
This theorem is referenced by:  eldju1st  6949
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