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Theorem djuin 9344
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 4163 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2 imassrn 5927 . . . 4 (inr “ 𝐵) ⊆ ran inr
3 djurf1o 9339 . . . . 5 inr:V–1-1-onto→({1o} × V)
4 f1of 6606 . . . . 5 (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5 frn 6509 . . . . 5 (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
63, 4, 5mp2b 10 . . . 4 ran inr ⊆ ({1o} × V)
72, 6sstri 3962 . . 3 (inr “ 𝐵) ⊆ ({1o} × V)
8 incom 4163 . . . 4 ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9 imassrn 5927 . . . . . 6 (inl “ 𝐴) ⊆ ran inl
10 djulf1o 9338 . . . . . . 7 inl:V–1-1-onto→({∅} × V)
11 f1of 6606 . . . . . . 7 (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12 frn 6509 . . . . . . 7 (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
1310, 11, 12mp2b 10 . . . . . 6 ran inl ⊆ ({∅} × V)
149, 13sstri 3962 . . . . 5 (inl “ 𝐴) ⊆ ({∅} × V)
15 1n0 8115 . . . . . . 7 1o ≠ ∅
1615necomi 3068 . . . . . 6 ∅ ≠ 1o
17 disjsn2 4633 . . . . . 6 (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18 xpdisj1 6005 . . . . . 6 (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
1916, 17, 18mp2b 10 . . . . 5 (({∅} × V) ∩ ({1o} × V)) = ∅
20 ssdisj 4392 . . . . 5 (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
2114, 19, 20mp2an 691 . . . 4 ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
228, 21eqtr3i 2849 . . 3 (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23 ssdisj 4392 . . 3 (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
247, 22, 23mp2an 691 . 2 ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
251, 24eqtr3i 2849 1 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wne 3014  Vcvv 3480  cin 3918  wss 3919  c0 4276  {csn 4550   × cxp 5540  ran crn 5543  cima 5545  wf 6339  1-1-ontowf1o 6342  1oc1o 8091  inlcinl 9325  inrcinr 9326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-om 7575  df-1st 7684  df-2nd 7685  df-1o 8098  df-inl 9328  df-inr 9329
This theorem is referenced by: (None)
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