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Theorem eldju1st 9992
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 9989 . 2 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 ssel2 4003 . . 3 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴𝐵)))
3 xp1st 8062 . . 3 (𝑋 ∈ ({∅, 1o} × (𝐴𝐵)) → (1st𝑋) ∈ {∅, 1o})
4 elpri 4671 . . 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 395  wo 846   = wceq 1537  wcel 2108  cun 3974  wss 3976  c0 4352  {cpr 4650   × cxp 5698  cfv 6573  1st c1st 8028  1oc1o 8515  cdju 9967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-suc 6401  df-iota 6525  df-fun 6575  df-fv 6581  df-1st 8030  df-2nd 8031  df-1o 8522  df-dju 9970  df-inl 9971  df-inr 9972
This theorem is referenced by: (None)
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