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Theorem djulclb 6906
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 6902 . 2 (𝐶𝐴 → (inl‘𝐶) ∈ (𝐴𝐵))
2 1n0 6295 . . . . . . . . . 10 1o ≠ ∅
32necomi 2368 . . . . . . . . 9 ∅ ≠ 1o
4 0ex 4023 . . . . . . . . . 10 ∅ ∈ V
54elsn 3511 . . . . . . . . 9 (∅ ∈ {1o} ↔ ∅ = 1o)
63, 5nemtbir 2372 . . . . . . . 8 ¬ ∅ ∈ {1o}
76intnanr 898 . . . . . . 7 ¬ (∅ ∈ {1o} ∧ 𝐶𝐵)
8 opelxp 4537 . . . . . . 7 (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ↔ (∅ ∈ {1o} ∧ 𝐶𝐵))
97, 8mtbir 643 . . . . . 6 ¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)
10 elex 2669 . . . . . . . . . . . 12 (𝐶𝑉𝐶 ∈ V)
11 opexg 4118 . . . . . . . . . . . . 13 ((∅ ∈ V ∧ 𝐶𝑉) → ⟨∅, 𝐶⟩ ∈ V)
124, 11mpan 418 . . . . . . . . . . . 12 (𝐶𝑉 → ⟨∅, 𝐶⟩ ∈ V)
13 opeq2 3674 . . . . . . . . . . . . 13 (𝑥 = 𝐶 → ⟨∅, 𝑥⟩ = ⟨∅, 𝐶⟩)
14 df-inl 6898 . . . . . . . . . . . . 13 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
1513, 14fvmptg 5463 . . . . . . . . . . . 12 ((𝐶 ∈ V ∧ ⟨∅, 𝐶⟩ ∈ V) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
1610, 12, 15syl2anc 406 . . . . . . . . . . 11 (𝐶𝑉 → (inl‘𝐶) = ⟨∅, 𝐶⟩)
1716adantr 272 . . . . . . . . . 10 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (inl‘𝐶) = ⟨∅, 𝐶⟩)
18 df-dju 6889 . . . . . . . . . . . . 13 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
1918eleq2i 2182 . . . . . . . . . . . 12 ((inl‘𝐶) ∈ (𝐴𝐵) ↔ (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2019biimpi 119 . . . . . . . . . . 11 ((inl‘𝐶) ∈ (𝐴𝐵) → (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2120adantl 273 . . . . . . . . . 10 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (inl‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
2217, 21eqeltrrd 2193 . . . . . . . . 9 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → ⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
23 elun 3185 . . . . . . . . 9 (⟨∅, 𝐶⟩ ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ∨ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)))
2422, 23sylib 121 . . . . . . . 8 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ∨ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵)))
2524orcomd 701 . . . . . . 7 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) ∨ ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
2625ord 696 . . . . . 6 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (¬ ⟨∅, 𝐶⟩ ∈ ({1o} × 𝐵) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴)))
279, 26mpi 15 . . . . 5 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → ⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴))
28 opelxp 4537 . . . . 5 (⟨∅, 𝐶⟩ ∈ ({∅} × 𝐴) ↔ (∅ ∈ {∅} ∧ 𝐶𝐴))
2927, 28sylib 121 . . . 4 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → (∅ ∈ {∅} ∧ 𝐶𝐴))
3029simprd 113 . . 3 ((𝐶𝑉 ∧ (inl‘𝐶) ∈ (𝐴𝐵)) → 𝐶𝐴)
3130ex 114 . 2 (𝐶𝑉 → ((inl‘𝐶) ∈ (𝐴𝐵) → 𝐶𝐴))
321, 31impbid2 142 1 (𝐶𝑉 → (𝐶𝐴 ↔ (inl‘𝐶) ∈ (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 680   = wceq 1314  wcel 1463  Vcvv 2658  cun 3037  c0 3331  {csn 3495  cop 3498   × cxp 4505  cfv 5091  1oc1o 6272  cdju 6888  inlcinl 6896
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-1o 6279  df-dju 6889  df-inl 6898
This theorem is referenced by:  exmidfodomrlemr  7022
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