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Theorem eldju1st 9913
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 9910 . 2 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 ssel2 3975 . . 3 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴𝐵)))
3 xp1st 8001 . . 3 (𝑋 ∈ ({∅, 1o} × (𝐴𝐵)) → (1st𝑋) ∈ {∅, 1o})
4 elpri 4648 . . 3 ((1st𝑋) ∈ {∅, 1o} → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
52, 3, 43syl 18 . 2 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
61, 5mpan 689 1 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  cun 3944  wss 3946  c0 4320  {cpr 4628   × cxp 5672  cfv 6539  1st c1st 7967  1oc1o 8453  cdju 9888
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 5297  ax-nul 5304  ax-pr 5425  ax-un 7719
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-suc 6366  df-iota 6491  df-fun 6541  df-fv 6547  df-1st 7969  df-2nd 7970  df-1o 8460  df-dju 9891  df-inl 9892  df-inr 9893
This theorem is referenced by: (None)
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