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Theorem djuin 6700
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 3181 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2 imassrn 4752 . . . 4 (inr “ 𝐵) ⊆ ran inr
3 djurf1o 6695 . . . . 5 inr:V–1-1-onto→({1𝑜} × V)
4 f1of 5216 . . . . 5 (inr:V–1-1-onto→({1𝑜} × V) → inr:V⟶({1𝑜} × V))
5 frn 5134 . . . . 5 (inr:V⟶({1𝑜} × V) → ran inr ⊆ ({1𝑜} × V))
63, 4, 5mp2b 8 . . . 4 ran inr ⊆ ({1𝑜} × V)
72, 6sstri 3023 . . 3 (inr “ 𝐵) ⊆ ({1𝑜} × V)
8 incom 3181 . . . 4 ((inl “ 𝐴) ∩ ({1𝑜} × V)) = (({1𝑜} × V) ∩ (inl “ 𝐴))
9 imassrn 4752 . . . . . 6 (inl “ 𝐴) ⊆ ran inl
10 djulf1o 6694 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
11 f1of 5216 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12 frn 5134 . . . . . . 7 (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
1310, 11, 12mp2b 8 . . . . . 6 ran inl ⊆ ({∅} × V)
149, 13sstri 3023 . . . . 5 (inl “ 𝐴) ⊆ ({∅} × V)
15 1n0 6151 . . . . . . 7 1𝑜 ≠ ∅
1615necomi 2336 . . . . . 6 ∅ ≠ 1𝑜
17 disjsn2 3488 . . . . . 6 (∅ ≠ 1𝑜 → ({∅} ∩ {1𝑜}) = ∅)
18 xpdisj1 4821 . . . . . 6 (({∅} ∩ {1𝑜}) = ∅ → (({∅} × V) ∩ ({1𝑜} × V)) = ∅)
1916, 17, 18mp2b 8 . . . . 5 (({∅} × V) ∩ ({1𝑜} × V)) = ∅
20 ssdisj 3327 . . . . 5 (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1𝑜} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1𝑜} × V)) = ∅)
2114, 19, 20mp2an 417 . . . 4 ((inl “ 𝐴) ∩ ({1𝑜} × V)) = ∅
228, 21eqtr3i 2107 . . 3 (({1𝑜} × V) ∩ (inl “ 𝐴)) = ∅
23 ssdisj 3327 . . 3 (((inr “ 𝐵) ⊆ ({1𝑜} × V) ∧ (({1𝑜} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
247, 22, 23mp2an 417 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
251, 24eqtr3i 2107 1 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1287  wne 2251  Vcvv 2615  cin 2987  wss 2988  c0 3275  {csn 3431   × cxp 4409  ran crn 4412  cima 4414  wf 4977  1-1-ontowf1o 4980  1𝑜c1o 6128  inlcinl 6681  inrcinr 6682
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-id 4094  df-iord 4167  df-on 4169  df-suc 4172  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-1st 5868  df-2nd 5869  df-1o 6135  df-inl 6683  df-inr 6684
This theorem is referenced by:  dju1p1e2  6767
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