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Theorem djuexALT 9863
Description: Alternate proof of djuex 9849, 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 5390 . . 3 {∅, 1o} ∈ V
2 unexg 7684 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3 xpexg 7685 . . 3 (({∅, 1o} ∈ V ∧ (𝐴𝐵) ∈ V) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
41, 2, 3sylancr 588 . 2 ((𝐴𝑉𝐵𝑊) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
5 djuss 9861 . . 3 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
65a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)))
74, 6ssexd 5282 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3444  cun 3909  wss 3911  c0 4283  {cpr 4589   × cxp 5632  1oc1o 8406  cdju 9839
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923  df-1o 8413  df-dju 9842  df-inl 9843  df-inr 9844
This theorem is referenced by: (None)
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