MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldju1st Structured version   Visualization version   GIF version

Theorem eldju1st 9946
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 9943 . 2 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 ssel2 3972 . . 3 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴𝐵)))
3 xp1st 8024 . . 3 (𝑋 ∈ ({∅, 1o} × (𝐴𝐵)) → (1st𝑋) ∈ {∅, 1o})
4 elpri 4652 . . 3 ((1st𝑋) ∈ {∅, 1o} → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
52, 3, 43syl 18 . 2 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
61, 5mpan 688 1 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  cun 3943  wss 3945  c0 4323  {cpr 4631   × cxp 5675  cfv 6547  1st c1st 7990  1oc1o 8478  cdju 9921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428  ax-un 7739
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6375  df-iota 6499  df-fun 6549  df-fv 6555  df-1st 7992  df-2nd 7993  df-1o 8485  df-dju 9924  df-inl 9925  df-inr 9926
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator
OSZAR »