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Theorem djuin 9910
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 ā€œ š“) ∩ (inr ā€œ šµ)) = āˆ…

Proof of Theorem djuin
StepHypRef Expression
1 incom 4194 . 2 ((inr ā€œ šµ) ∩ (inl ā€œ š“)) = ((inl ā€œ š“) ∩ (inr ā€œ šµ))
2 imassrn 6061 . . . 4 (inr ā€œ šµ) āŠ† ran inr
3 djurf1o 9905 . . . . 5 inr:V–1-1-onto→({1o} Ɨ V)
4 f1of 6824 . . . . 5 (inr:V–1-1-onto→({1o} Ɨ V) → inr:V⟶({1o} Ɨ V))
5 frn 6715 . . . . 5 (inr:V⟶({1o} Ɨ V) → ran inr āŠ† ({1o} Ɨ V))
63, 4, 5mp2b 10 . . . 4 ran inr āŠ† ({1o} Ɨ V)
72, 6sstri 3984 . . 3 (inr ā€œ šµ) āŠ† ({1o} Ɨ V)
8 incom 4194 . . . 4 ((inl ā€œ š“) ∩ ({1o} Ɨ V)) = (({1o} Ɨ V) ∩ (inl ā€œ š“))
9 imassrn 6061 . . . . . 6 (inl ā€œ š“) āŠ† ran inl
10 djulf1o 9904 . . . . . . 7 inl:V–1-1-onto→({āˆ…} Ɨ V)
11 f1of 6824 . . . . . . 7 (inl:V–1-1-onto→({āˆ…} Ɨ V) → inl:V⟶({āˆ…} Ɨ V))
12 frn 6715 . . . . . . 7 (inl:V⟶({āˆ…} Ɨ V) → ran inl āŠ† ({āˆ…} Ɨ V))
1310, 11, 12mp2b 10 . . . . . 6 ran inl āŠ† ({āˆ…} Ɨ V)
149, 13sstri 3984 . . . . 5 (inl ā€œ š“) āŠ† ({āˆ…} Ɨ V)
15 1n0 8484 . . . . . . 7 1o ≠ āˆ…
1615necomi 2987 . . . . . 6 āˆ… ≠ 1o
17 disjsn2 4709 . . . . . 6 (āˆ… ≠ 1o → ({āˆ…} ∩ {1o}) = āˆ…)
18 xpdisj1 6151 . . . . . 6 (({āˆ…} ∩ {1o}) = āˆ… → (({āˆ…} Ɨ V) ∩ ({1o} Ɨ V)) = āˆ…)
1916, 17, 18mp2b 10 . . . . 5 (({āˆ…} Ɨ V) ∩ ({1o} Ɨ V)) = āˆ…
20 ssdisj 4452 . . . . 5 (((inl ā€œ š“) āŠ† ({āˆ…} Ɨ V) ∧ (({āˆ…} Ɨ V) ∩ ({1o} Ɨ V)) = āˆ…) → ((inl ā€œ š“) ∩ ({1o} Ɨ V)) = āˆ…)
2114, 19, 20mp2an 689 . . . 4 ((inl ā€œ š“) ∩ ({1o} Ɨ V)) = āˆ…
228, 21eqtr3i 2754 . . 3 (({1o} Ɨ V) ∩ (inl ā€œ š“)) = āˆ…
23 ssdisj 4452 . . 3 (((inr ā€œ šµ) āŠ† ({1o} Ɨ V) ∧ (({1o} Ɨ V) ∩ (inl ā€œ š“)) = āˆ…) → ((inr ā€œ šµ) ∩ (inl ā€œ š“)) = āˆ…)
247, 22, 23mp2an 689 . 2 ((inr ā€œ šµ) ∩ (inl ā€œ š“)) = āˆ…
251, 24eqtr3i 2754 1 ((inl ā€œ š“) ∩ (inr ā€œ šµ)) = āˆ…
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   ≠ wne 2932  Vcvv 3466   ∩ cin 3940   āŠ† wss 3941  āˆ…c0 4315  {csn 4621   Ɨ cxp 5665  ran crn 5668   ā€œ cima 5670  āŸ¶wf 6530  ā€“1-1-onto→wf1o 6533  1oc1o 8455  inlcinl 9891  inrcinr 9892
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 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-om 7850  df-1st 7969  df-2nd 7970  df-1o 8462  df-inl 9894  df-inr 9895
This theorem is referenced by: (None)
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