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Theorem eldju2ndl 6907
 Description: The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)
Assertion
Ref Expression
eldju2ndl ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)

Proof of Theorem eldju2ndl
StepHypRef Expression
1 df-dju 6873 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2179 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 3181 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 183 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 6019 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 simprr 504 . . . . . 6 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → (2nd𝑋) ∈ 𝐴)
76a1d 22 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
85, 7sylbi 120 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
9 elxp6 6019 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
10 elsni 3509 . . . . . . 7 ((1st𝑋) ∈ {1o} → (1st𝑋) = 1o)
11 1n0 6281 . . . . . . . 8 1o ≠ ∅
12 neeq1 2293 . . . . . . . 8 ((1st𝑋) = 1o → ((1st𝑋) ≠ ∅ ↔ 1o ≠ ∅))
1311, 12mpbiri 167 . . . . . . 7 ((1st𝑋) = 1o → (1st𝑋) ≠ ∅)
14 eqneqall 2290 . . . . . . . 8 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐴))
1514com12 30 . . . . . . 7 ((1st𝑋) ≠ ∅ → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1610, 13, 153syl 17 . . . . . 6 ((1st𝑋) ∈ {1o} → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1716ad2antrl 479 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
189, 17sylbi 120 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
198, 18jaoi 688 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
204, 19sylbi 120 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
2120imp 123 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 680   = wceq 1312   ∈ wcel 1461   ≠ wne 2280   ∪ cun 3033  ∅c0 3327  {csn 3491  ⟨cop 3494   × cxp 4495  ‘cfv 5079  1st c1st 5988  2nd c2nd 5989  1oc1o 6258   ⊔ cdju 6872 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313 This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-iota 5044  df-fun 5081  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-dju 6873 This theorem is referenced by:  updjudhf  6914
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