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Theorem eldju2ndl 9919
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 9896 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2826 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 4149 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 275 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 8009 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 simprr 772 . . . . . 6 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → (2nd𝑋) ∈ 𝐴)
76a1d 25 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
85, 7sylbi 216 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
9 elxp6 8009 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
10 elsni 4646 . . . . . . 7 ((1st𝑋) ∈ {1o} → (1st𝑋) = 1o)
11 1n0 8488 . . . . . . . 8 1o ≠ ∅
12 neeq1 3004 . . . . . . . 8 ((1st𝑋) = 1o → ((1st𝑋) ≠ ∅ ↔ 1o ≠ ∅))
1311, 12mpbiri 258 . . . . . . 7 ((1st𝑋) = 1o → (1st𝑋) ≠ ∅)
14 eqneqall 2952 . . . . . . . 8 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐴))
1514com12 32 . . . . . . 7 ((1st𝑋) ≠ ∅ → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1610, 13, 153syl 18 . . . . . 6 ((1st𝑋) ∈ {1o} → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1716ad2antrl 727 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
189, 17sylbi 216 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
198, 18jaoi 856 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
204, 19sylbi 216 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
2120imp 408 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  cun 3947  c0 4323  {csn 4629  cop 4635   × cxp 5675  cfv 6544  1st c1st 7973  2nd c2nd 7974  1oc1o 8459  cdju 9893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976  df-1o 8466  df-dju 9896
This theorem is referenced by:  updjudhf  9926
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