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Theorem djuexALT 9919
Description: Alternate proof of djuex 9905, 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 5431 . . 3 {∅, 1o} ∈ V
2 unexg 7738 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3 xpexg 7739 . . 3 (({∅, 1o} ∈ V ∧ (𝐴𝐵) ∈ V) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
41, 2, 3sylancr 585 . 2 ((𝐴𝑉𝐵𝑊) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
5 djuss 9917 . . 3 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
65a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)))
74, 6ssexd 5323 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  Vcvv 3472  cun 3945  wss 3947  c0 4321  {cpr 4629   × cxp 5673  1oc1o 8461  cdju 9895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-suc 6369  df-iota 6494  df-fun 6544  df-fv 6550  df-1st 7977  df-2nd 7978  df-1o 8468  df-dju 9898  df-inl 9899  df-inr 9900
This theorem is referenced by: (None)
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