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Theorem eldju2ndr 9922
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 9898 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2819 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 4143 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 275 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 8008 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 elsni 4640 . . . . . . 7 ((1st𝑋) ∈ {∅} → (1st𝑋) = ∅)
7 eqneqall 2945 . . . . . . 7 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
86, 7syl 17 . . . . . 6 ((1st𝑋) ∈ {∅} → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
98ad2antrl 725 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
105, 9sylbi 216 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
11 elxp6 8008 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
12 simprr 770 . . . . . 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 216 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
1510, 14jaoi 854 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐵))
164, 15sylbi 216 . 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 844   = wceq 1533  wcel 2098  wne 2934  cun 3941  c0 4317  {csn 4623  cop 4629   × cxp 5667  cfv 6537  1st c1st 7972  2nd c2nd 7973  1oc1o 8460  cdju 9895
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-1st 7974  df-2nd 7975  df-dju 9898
This theorem is referenced by:  updjudhf  9928
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