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Theorem djuexALT 9917
Description: Alternate proof of djuex 9903, which is shorter, but based indirectly on the definitions of inl and inr. (Proposed by BJ, 28-Jun-2022.) (Contributed by AV, 28-Jun-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
djuexALT ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem djuexALT
StepHypRef Expression
1 prex 5433 . . 3 {∅, 1o} ∈ V
2 unexg 7736 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3 xpexg 7737 . . 3 (({∅, 1o} ∈ V ∧ (𝐴𝐵) ∈ V) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
41, 2, 3sylancr 588 . 2 ((𝐴𝑉𝐵𝑊) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
5 djuss 9915 . . 3 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
65a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)))
74, 6ssexd 5325 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3475  cun 3947  wss 3949  c0 4323  {cpr 4631   × cxp 5675  1oc1o 8459  cdju 9893
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-pow 5364  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-pw 4605  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: (None)
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