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Theorem eldju2ndr 9963
Description: The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)
Assertion
Ref Expression
eldju2ndr ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) ≠ ∅) → (2nd𝑋) ∈ 𝐵)

Proof of Theorem eldju2ndr
StepHypRef Expression
1 df-dju 9939 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2831 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 4163 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 275 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 8047 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 elsni 4648 . . . . . . 7 ((1st𝑋) ∈ {∅} → (1st𝑋) = ∅)
7 eqneqall 2949 . . . . . . 7 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
86, 7syl 17 . . . . . 6 ((1st𝑋) ∈ {∅} → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
98ad2antrl 728 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
105, 9sylbi 217 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
11 elxp6 8047 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
12 simprr 773 . . . . . 6 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → (2nd𝑋) ∈ 𝐵)
1312a1d 25 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1411, 13sylbi 217 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1510, 14jaoi 857 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
164, 15sylbi 217 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1716imp 406 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) ≠ ∅) → (2nd𝑋) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  cun 3961  c0 4339  {csn 4631  cop 4637   × cxp 5687  cfv 6563  1st c1st 8011  2nd c2nd 8012  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-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014  df-dju 9939
This theorem is referenced by:  updjudhf  9969
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