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Theorem djuin 9909
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 4193 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2 imassrn 6060 . . . 4 (inr “ 𝐵) ⊆ ran inr
3 djurf1o 9904 . . . . 5 inr:V–1-1-onto→({1o} × V)
4 f1of 6823 . . . . 5 (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5 frn 6714 . . . . 5 (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
63, 4, 5mp2b 10 . . . 4 ran inr ⊆ ({1o} × V)
72, 6sstri 3983 . . 3 (inr “ 𝐵) ⊆ ({1o} × V)
8 incom 4193 . . . 4 ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9 imassrn 6060 . . . . . 6 (inl “ 𝐴) ⊆ ran inl
10 djulf1o 9903 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
11 f1of 6823 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12 frn 6714 . . . . . . 7 (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
1310, 11, 12mp2b 10 . . . . . 6 ran inl ⊆ ({∅} × V)
149, 13sstri 3983 . . . . 5 (inl “ 𝐴) ⊆ ({∅} × V)
15 1n0 8483 . . . . . . 7 1o ≠ ∅
1615necomi 2987 . . . . . 6 ∅ ≠ 1o
17 disjsn2 4708 . . . . . 6 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18 xpdisj1 6150 . . . . . 6 (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
1916, 17, 18mp2b 10 . . . . 5 (({∅} × V) ∩ ({1o} × V)) = ∅
20 ssdisj 4451 . . . . 5 (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
2114, 19, 20mp2an 689 . . . 4 ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
228, 21eqtr3i 2754 . . 3 (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23 ssdisj 4451 . . 3 (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
247, 22, 23mp2an 689 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
251, 24eqtr3i 2754 1 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wne 2932  Vcvv 3466  cin 3939  wss 3940  c0 4314  {csn 4620   × cxp 5664  ran crn 5667  cima 5669  wf 6529  1-1-ontowf1o 6532  1oc1o 8454  inlcinl 9890  inrcinr 9891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7849  df-1st 7968  df-2nd 7969  df-1o 8461  df-inl 9893  df-inr 9894
This theorem is referenced by: (None)
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