MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eldju2ndl Structured version   Visualization version   GIF version

Theorem eldju2ndl 9964
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 9941 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2833 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 4153 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 275 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 8048 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 simprr 773 . . . . . 6 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → (2nd𝑋) ∈ 𝐴)
76a1d 25 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
85, 7sylbi 217 . . . 4 (𝑋 ∈ ({∅} × 𝐴) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
9 elxp6 8048 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
10 elsni 4643 . . . . . . 7 ((1st𝑋) ∈ {1o} → (1st𝑋) = 1o)
11 1n0 8526 . . . . . . . 8 1o ≠ ∅
12 neeq1 3003 . . . . . . . 8 ((1st𝑋) = 1o → ((1st𝑋) ≠ ∅ ↔ 1o ≠ ∅))
1311, 12mpbiri 258 . . . . . . 7 ((1st𝑋) = 1o → (1st𝑋) ≠ ∅)
14 eqneqall 2951 . . . . . . . 8 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐴))
1514com12 32 . . . . . . 7 ((1st𝑋) ≠ ∅ → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1610, 13, 153syl 18 . . . . . 6 ((1st𝑋) ∈ {1o} → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1716ad2antrl 728 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
189, 17sylbi 217 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
198, 18jaoi 858 . . 3 ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
204, 19sylbi 217 . 2 (𝑋 ∈ (𝐴𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
2120imp 406 1 ((𝑋 ∈ (𝐴𝐵) ∧ (1st𝑋) = ∅) → (2nd𝑋) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  cun 3949  c0 4333  {csn 4626  cop 4632   × cxp 5683  cfv 6561  1st c1st 8012  2nd c2nd 8013  1oc1o 8499  cdju 9938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015  df-1o 8506  df-dju 9941
This theorem is referenced by:  updjudhf  9971
  Copyright terms: Public domain W3C validator