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Theorem eldju1st 9961
Description: The first component of an element of a disjoint union is either or 1o. (Contributed by AV, 26-Jun-2022.)
Assertion
Ref Expression
eldju1st (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))

Proof of Theorem eldju1st
StepHypRef Expression
1 djuss 9958 . 2 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 ssel2 3990 . . 3 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴𝐵)))
3 xp1st 8045 . . 3 (𝑋 ∈ ({∅, 1o} × (𝐴𝐵)) → (1st𝑋) ∈ {∅, 1o})
4 elpri 4654 . . 3 ((1st𝑋) ∈ {∅, 1o} → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
52, 3, 43syl 18 . 2 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
61, 5mpan 690 1 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  cun 3961  wss 3963  c0 4339  {cpr 4633   × cxp 5687  cfv 6563  1st c1st 8011  1oc1o 8498  cdju 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-1o 8505  df-dju 9939  df-inl 9940  df-inr 9941
This theorem is referenced by: (None)
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