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Theorem djulclb 7048
Description: Left biconditional closure of disjoint union. (Contributed by Jim Kingdon, 2-Jul-2022.)
Assertion
Ref Expression
djulclb (𝐶𝑉 → (𝐶𝐴 ↔ (inl‘𝐶) ∈ (𝐴𝐵)))

Proof of Theorem djulclb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 djulcl 7044 . 2 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
2 1n0 6427 . . . . . . . . . 10 1o ≠ ∅
32necomi 2432 . . . . . . . . 9 ∅ ≠ 1o
4 0ex 4127 . . . . . . . . . 10 ∅ ∈ V
54elsn 3607 . . . . . . . . 9 (∅ ∈ {1o} ↔ ∅ = 1o)
63, 5nemtbir 2436 . . . . . . . 8 ¬ ∅ ∈ {1o}
76intnanr 930 . . . . . . 7 ¬ (∅ ∈ {1o} ∧ 𝐶𝐵)
8 opelxp 4653 . . . . . . 7 (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ↔ (∅ ∈ {1o} ∧ 𝐶𝐵))
97, 8mtbir 671 . . . . . 6 ¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)
10 elex 2748 . . . . . . . . . . . 12 (𝐶𝑉𝐶 ∈ V)
11 opexg 4225 . . . . . . . . . . . . 13 ((∅ ∈ V ∧ 𝐶𝑉) → ⟨∅, 𝐶⟩ ∈ V)
124, 11mpan 424 . . . . . . . . . . . 12 (𝐶𝑉 → ⟨∅, 𝐶⟩ ∈ V)
13 opeq2 3777 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
14 df-inl 7040 . . . . . . . . . . . . 13 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
1513, 14fvmptg 5588 . . . . . . . . . . . 12 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ V) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
1610, 12, 15syl2anc 411 . . . . . . . . . . 11 (𝐶𝑉 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
1716adantr 276 . . . . . . . . . 10 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
18 df-dju 7031 . . . . . . . . . . . . 13 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1918eleq2i 2244 . . . . . . . . . . . 12 ((inl‘𝐶) ∈ (𝐴𝐵) ↔ (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2019biimpi 120 . . . . . . . . . . 11 ((inl‘𝐶) ∈ (𝐴𝐵) → (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2120adantl 277 . . . . . . . . . 10 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2217, 21eqeltrrd 2255 . . . . . . . . 9 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
23 elun 3276 . . . . . . . . 9 (⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ∨ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)))
2422, 23sylib 122 . . . . . . . 8 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ∨ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)))
2524orcomd 729 . . . . . . 7 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ∨ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
2625ord 724 . . . . . 6 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
279, 26mpi 15 . . . . 5 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
28 opelxp 4653 . . . . 5 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ↔ (∅ ∈ {∅} ∧ 𝐶𝐴))
2927, 28sylib 122 . . . 4 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (∅ ∈ {∅} ∧ 𝐶𝐴))
3029simprd 114 . . 3 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → 𝐶𝐴)
3130ex 115 . 2 (𝐶𝑉 → ((inl‘𝐶) ∈ (𝐴𝐵) → 𝐶𝐴))
321, 31impbid2 143 1 (𝐶𝑉 → (𝐶𝐴 ↔ (inl‘𝐶) ∈ (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127  c0 3422  {csn 3591  cop 3594   × cxp 4621  cfv 5212  1oc1o 6404  cdju 7030  inlcinl 7038
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-1o 6411  df-dju 7031  df-inl 7040
This theorem is referenced by:  exmidfodomrlemr  7195
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