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Theorem eldju2ndr 6962
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 6927 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2207 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 3218 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 183 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 6071 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 elsni 3546 . . . . . . 7 ((1st𝑋) ∈ {∅} → (1st𝑋) = ∅)
7 eqneqall 2319 . . . . . . 7 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
86, 7syl 14 . . . . . 6 ((1st𝑋) ∈ {∅} → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
98ad2antrl 482 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
105, 9sylbi 120 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
11 elxp6 6071 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
12 simprr 522 . . . . . 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 120 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1510, 14jaoi 706 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
164, 15sylbi 120 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1716imp 123 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) ≠ ∅) → (2nd𝑋) ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 698   = wceq 1332  wcel 1481  wne 2309  cun 3070  c0 3364  {csn 3528  cop 3531   × cxp 4541  cfv 5127  1st c1st 6040  2nd c2nd 6041  1oc1o 6310  cdju 6926
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-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-iota 5092  df-fun 5129  df-fv 5135  df-1st 6042  df-2nd 6043  df-dju 6927
This theorem is referenced by:  updjudhf  6968
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