MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  djuexALT Structured version   Visualization version   GIF version

Theorem djuexALT 9916
Description: Alternate proof of djuex 9902, 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 5432 . . 3 {∅, 1o} ∈ V
2 unexg 7735 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3 xpexg 7736 . . 3 (({∅, 1o} ∈ V ∧ (𝐴𝐵) ∈ V) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
41, 2, 3sylancr 587 . 2 ((𝐴𝑉𝐵𝑊) → ({∅, 1o} × (𝐴𝐵)) ∈ V)
5 djuss 9914 . . 3 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
65a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)))
74, 6ssexd 5324 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3474  cun 3946  wss 3948  c0 4322  {cpr 4630   × cxp 5674  1oc1o 8458  cdju 9892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-suc 6370  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7974  df-2nd 7975  df-1o 8465  df-dju 9895  df-inl 9896  df-inr 9897
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator