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Theorem eldju2ndl 7049
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 7015 . . . . 5 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2237 . . . 4 (𝑋 ∈ (𝐴𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 3268 . . . 4 (𝑋 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
42, 3bitri 183 . . 3 (𝑋 ∈ (𝐴𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1o} × 𝐵)))
5 elxp6 6148 . . . . 5 (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {∅} ∧ (2nd𝑋) ∈ 𝐴)))
6 simprr 527 . . . . . 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 6148 . . . . 5 (𝑋 ∈ ({1o} × 𝐵) ↔ (𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)))
10 elsni 3601 . . . . . . 7 ((1st𝑋) ∈ {1o} → (1st𝑋) = 1o)
11 1n0 6411 . . . . . . . 8 1o ≠ ∅
12 neeq1 2353 . . . . . . . 8 ((1st𝑋) = 1o → ((1st𝑋) ≠ ∅ ↔ 1o ≠ ∅))
1311, 12mpbiri 167 . . . . . . 7 ((1st𝑋) = 1o → (1st𝑋) ≠ ∅)
14 eqneqall 2350 . . . . . . . 8 ((1st𝑋) = ∅ → ((1st𝑋) ≠ ∅ → (2nd𝑋) ∈ 𝐴))
1514com12 30 . . . . . . 7 ((1st𝑋) ≠ ∅ → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1610, 13, 153syl 17 . . . . . 6 ((1st𝑋) ∈ {1o} → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
1716ad2antrl 487 . . . . 5 ((𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩ ∧ ((1st𝑋) ∈ {1o} ∧ (2nd𝑋) ∈ 𝐵)) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
189, 17sylbi 120 . . . 4 (𝑋 ∈ ({1o} × 𝐵) → ((1st𝑋) = ∅ → (2nd𝑋) ∈ 𝐴))
198, 18jaoi 711 . . 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 703   = wceq 1348  wcel 2141  wne 2340  cun 3119  c0 3414  {csn 3583  cop 3586   × cxp 4609  cfv 5198  1st c1st 6117  2nd c2nd 6118  1oc1o 6388  cdju 7014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015
This theorem is referenced by:  updjudhf  7056  subctctexmid  14034
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