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Theorem eldju2ndl 7084
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 7050 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2254 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 3288 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 184 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 6183 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 simprr 531 . . . . . 6 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → (2nd𝑋) ∈ 𝐴)
76a1d 22 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
85, 7sylbi 121 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
9 elxp6 6183 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
10 elsni 3622 . . . . . . 7 ((1st𝑋) ∈ {1o} → (1st𝑋) = 1o)
11 1n0 6446 . . . . . . . 8 1o ≠ ∅
12 neeq1 2370 . . . . . . . 8 ((1st𝑋) = 1o → ((1st𝑋) ≠ ∅ ↔ 1o ≠ ∅))
1311, 12mpbiri 168 . . . . . . 7 ((1st𝑋) = 1o → (1st𝑋) ≠ ∅)
14 eqneqall 2367 . . . . . . . 8 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐴))
1514com12 30 . . . . . . 7 ((1st𝑋) ≠ ∅ → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1610, 13, 153syl 17 . . . . . 6 ((1st𝑋) ∈ {1o} → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1716ad2antrl 490 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
189, 17sylbi 121 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
198, 18jaoi 717 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
204, 19sylbi 121 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
2120imp 124 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1363  wcel 2158  wne 2357  cun 3139  c0 3434  {csn 3604  cop 3607   × cxp 4636  cfv 5228  1st c1st 6152  2nd c2nd 6153  1oc1o 6423  cdju 7049
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-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-iota 5190  df-fun 5230  df-fv 5236  df-1st 6154  df-2nd 6155  df-1o 6430  df-dju 7050
This theorem is referenced by:  updjudhf  7091  subctctexmid  14991
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