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Theorem eldju2ndr 7091
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 7056 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2256 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 3291 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 184 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 6188 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 elsni 3625 . . . . . . 7 ((1st𝑋) ∈ {∅} → (1st𝑋) = ∅)
7 eqneqall 2370 . . . . . . 7 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
86, 7syl 14 . . . . . 6 ((1st𝑋) ∈ {∅} → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
98ad2antrl 490 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
105, 9sylbi 121 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
11 elxp6 6188 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
12 simprr 531 . . . . . 6 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → (2nd𝑋) ∈ 𝐵)
1312a1d 22 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1411, 13sylbi 121 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1510, 14jaoi 717 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
164, 15sylbi 121 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1716imp 124 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) ≠ ∅) → (2nd𝑋) ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wcel 2160  wne 2360  cun 3142  c0 3437  {csn 3607  cop 3610   × cxp 4639  cfv 5231  1st c1st 6157  2nd c2nd 6158  1oc1o 6428  cdju 7055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5233  df-fv 5239  df-1st 6159  df-2nd 6160  df-dju 7056
This theorem is referenced by:  updjudhf  7097
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