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Theorem djuss 9915
Description: A disjoint union is a subclass of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
Assertion
Ref Expression
djuss (𝐴 βŠ” 𝐡) βŠ† ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡))

Proof of Theorem djuss
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djur 9914 . . 3 (π‘₯ ∈ (𝐴 βŠ” 𝐡) β†’ (βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (inlβ€˜π‘¦) ∨ βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (inrβ€˜π‘¦)))
2 simpr 486 . . . . . . 7 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ π‘₯ = (inlβ€˜π‘¦))
3 df-inl 9897 . . . . . . . . 9 inl = (π‘₯ ∈ V ↦ βŸ¨βˆ…, π‘₯⟩)
4 opeq2 4875 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ βŸ¨βˆ…, π‘₯⟩ = βŸ¨βˆ…, π‘¦βŸ©)
5 elex 3493 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ V)
6 opex 5465 . . . . . . . . . 10 βŸ¨βˆ…, π‘¦βŸ© ∈ V
76a1i 11 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ βŸ¨βˆ…, π‘¦βŸ© ∈ V)
83, 4, 5, 7fvmptd3 7022 . . . . . . . 8 (𝑦 ∈ 𝐴 β†’ (inlβ€˜π‘¦) = βŸ¨βˆ…, π‘¦βŸ©)
98adantr 482 . . . . . . 7 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ (inlβ€˜π‘¦) = βŸ¨βˆ…, π‘¦βŸ©)
102, 9eqtrd 2773 . . . . . 6 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ π‘₯ = βŸ¨βˆ…, π‘¦βŸ©)
11 elun1 4177 . . . . . . . . 9 (𝑦 ∈ 𝐴 β†’ 𝑦 ∈ (𝐴 βˆͺ 𝐡))
12 0ex 5308 . . . . . . . . . 10 βˆ… ∈ V
1312prid1 4767 . . . . . . . . 9 βˆ… ∈ {βˆ…, 1o}
1411, 13jctil 521 . . . . . . . 8 (𝑦 ∈ 𝐴 β†’ (βˆ… ∈ {βˆ…, 1o} ∧ 𝑦 ∈ (𝐴 βˆͺ 𝐡)))
1514adantr 482 . . . . . . 7 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ (βˆ… ∈ {βˆ…, 1o} ∧ 𝑦 ∈ (𝐴 βˆͺ 𝐡)))
16 opelxp 5713 . . . . . . 7 (βŸ¨βˆ…, π‘¦βŸ© ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)) ↔ (βˆ… ∈ {βˆ…, 1o} ∧ 𝑦 ∈ (𝐴 βˆͺ 𝐡)))
1715, 16sylibr 233 . . . . . 6 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ βŸ¨βˆ…, π‘¦βŸ© ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
1810, 17eqeltrd 2834 . . . . 5 ((𝑦 ∈ 𝐴 ∧ π‘₯ = (inlβ€˜π‘¦)) β†’ π‘₯ ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
1918rexlimiva 3148 . . . 4 (βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (inlβ€˜π‘¦) β†’ π‘₯ ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
20 simpr 486 . . . . . . 7 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ π‘₯ = (inrβ€˜π‘¦))
21 df-inr 9898 . . . . . . . . 9 inr = (π‘₯ ∈ V ↦ ⟨1o, π‘₯⟩)
22 opeq2 4875 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ ⟨1o, π‘₯⟩ = ⟨1o, π‘¦βŸ©)
23 elex 3493 . . . . . . . . 9 (𝑦 ∈ 𝐡 β†’ 𝑦 ∈ V)
24 opex 5465 . . . . . . . . . 10 ⟨1o, π‘¦βŸ© ∈ V
2524a1i 11 . . . . . . . . 9 (𝑦 ∈ 𝐡 β†’ ⟨1o, π‘¦βŸ© ∈ V)
2621, 22, 23, 25fvmptd3 7022 . . . . . . . 8 (𝑦 ∈ 𝐡 β†’ (inrβ€˜π‘¦) = ⟨1o, π‘¦βŸ©)
2726adantr 482 . . . . . . 7 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ (inrβ€˜π‘¦) = ⟨1o, π‘¦βŸ©)
2820, 27eqtrd 2773 . . . . . 6 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ π‘₯ = ⟨1o, π‘¦βŸ©)
29 elun2 4178 . . . . . . . . 9 (𝑦 ∈ 𝐡 β†’ 𝑦 ∈ (𝐴 βˆͺ 𝐡))
3029adantr 482 . . . . . . . 8 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ 𝑦 ∈ (𝐴 βˆͺ 𝐡))
31 1oex 8476 . . . . . . . . 9 1o ∈ V
3231prid2 4768 . . . . . . . 8 1o ∈ {βˆ…, 1o}
3330, 32jctil 521 . . . . . . 7 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ (1o ∈ {βˆ…, 1o} ∧ 𝑦 ∈ (𝐴 βˆͺ 𝐡)))
34 opelxp 5713 . . . . . . 7 (⟨1o, π‘¦βŸ© ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)) ↔ (1o ∈ {βˆ…, 1o} ∧ 𝑦 ∈ (𝐴 βˆͺ 𝐡)))
3533, 34sylibr 233 . . . . . 6 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ ⟨1o, π‘¦βŸ© ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
3628, 35eqeltrd 2834 . . . . 5 ((𝑦 ∈ 𝐡 ∧ π‘₯ = (inrβ€˜π‘¦)) β†’ π‘₯ ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
3736rexlimiva 3148 . . . 4 (βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (inrβ€˜π‘¦) β†’ π‘₯ ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
3819, 37jaoi 856 . . 3 ((βˆƒπ‘¦ ∈ 𝐴 π‘₯ = (inlβ€˜π‘¦) ∨ βˆƒπ‘¦ ∈ 𝐡 π‘₯ = (inrβ€˜π‘¦)) β†’ π‘₯ ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
391, 38syl 17 . 2 (π‘₯ ∈ (𝐴 βŠ” 𝐡) β†’ π‘₯ ∈ ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡)))
4039ssriv 3987 1 (𝐴 βŠ” 𝐡) βŠ† ({βˆ…, 1o} Γ— (𝐴 βˆͺ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  {cpr 4631  βŸ¨cop 4635   Γ— cxp 5675  β€˜cfv 6544  1oc1o 8459   βŠ” cdju 9893  inlcinl 9894  inrcinr 9895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976  df-1o 8466  df-dju 9896  df-inl 9897  df-inr 9898
This theorem is referenced by:  djuunxp  9916  djuexALT  9917  eldju1st  9918
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