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Theorem eldju1st 7046
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 7045 . 2 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
2 ssel2 3142 . . 3 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → 𝑋 ∈ ({∅, 1o} × (𝐴𝐵)))
3 xp1st 6142 . . 3 (𝑋 ∈ ({∅, 1o} × (𝐴𝐵)) → (1st𝑋) ∈ {∅, 1o})
4 elpri 3604 . . 3 ((1st𝑋) ∈ {∅, 1o} → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
52, 3, 43syl 17 . 2 (((𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵)) ∧ 𝑋 ∈ (𝐴𝐵)) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
61, 5mpan 422 1 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ ∨ (1st𝑋) = 1o))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 703   = wceq 1348  wcel 2141  cun 3119  wss 3121  c0 3414  {cpr 3582   × cxp 4607  cfv 5196  1st c1st 6115  1oc1o 6386  cdju 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6117  df-2nd 6118  df-1o 6393  df-dju 7013  df-inl 7022  df-inr 7023
This theorem is referenced by:  updjudhf  7054  subctctexmid  13999
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