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Theorem djuin 6663
 Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)
Assertion
Ref Expression
djuin ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Proof of Theorem djuin
StepHypRef Expression
1 incom 3176 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2 imassrn 4740 . . . 4 (inr “ 𝐵) ⊆ ran inr
3 djurf1o 6658 . . . . 5 inr:V–1-1-onto→({1𝑜} × V)
4 f1of 5201 . . . . 5 (inr:V–1-1-onto→({1𝑜} × V) → inr:V⟶({1𝑜} × V))
5 frn 5121 . . . . 5 (inr:V⟶({1𝑜} × V) → ran inr ⊆ ({1𝑜} × V))
63, 4, 5mp2b 8 . . . 4 ran inr ⊆ ({1𝑜} × V)
72, 6sstri 3019 . . 3 (inr “ 𝐵) ⊆ ({1𝑜} × V)
8 incom 3176 . . . 4 ((inl “ 𝐴) ∩ ({1𝑜} × V)) = (({1𝑜} × V) ∩ (inl “ 𝐴))
9 imassrn 4740 . . . . . 6 (inl “ 𝐴) ⊆ ran inl
10 djulf1o 6657 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
11 f1of 5201 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12 frn 5121 . . . . . . 7 (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
1310, 11, 12mp2b 8 . . . . . 6 ran inl ⊆ ({∅} × V)
149, 13sstri 3019 . . . . 5 (inl “ 𝐴) ⊆ ({∅} × V)
15 1n0 6129 . . . . . . 7 1𝑜 ≠ ∅
1615necomi 2334 . . . . . 6 ∅ ≠ 1𝑜
17 disjsn2 3479 . . . . . 6 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
18 xpdisj1 4809 . . . . . 6 (({∅} ∩ {1𝑜}) = ∅ → (({∅} × V) ∩ ({1𝑜} × V)) = ∅)
1916, 17, 18mp2b 8 . . . . 5 (({∅} × V) ∩ ({1𝑜} × V)) = ∅
20 ssdisj 3321 . . . . 5 (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1𝑜} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1𝑜} × V)) = ∅)
2114, 19, 20mp2an 417 . . . 4 ((inl “ 𝐴) ∩ ({1𝑜} × V)) = ∅
228, 21eqtr3i 2105 . . 3 (({1𝑜} × V) ∩ (inl “ 𝐴)) = ∅
23 ssdisj 3321 . . 3 (((inr “ 𝐵) ⊆ ({1𝑜} × V) ∧ (({1𝑜} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
247, 22, 23mp2an 417 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
251, 24eqtr3i 2105 1 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1285   ≠ wne 2249  Vcvv 2612   ∩ cin 2983   ⊆ wss 2984  ∅c0 3269  {csn 3422   × cxp 4399  ran crn 4402   “ cima 4404  ⟶wf 4965  –1-1-onto→wf1o 4968  1𝑜c1o 6106  inlcinl 6644  inrcinr 6645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-1st 5846  df-2nd 5847  df-1o 6113  df-inl 6646  df-inr 6647 This theorem is referenced by:  dju1p1e2  6726
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