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Theorem djuin 9861
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“) āˆ© (inr ā€œ šµ)) = āˆ…

Proof of Theorem djuin
StepHypRef Expression
1 incom 4166 . 2 ((inr ā€œ šµ) āˆ© (inl ā€œ š“)) = ((inl ā€œ š“) āˆ© (inr ā€œ šµ))
2 imassrn 6029 . . . 4 (inr ā€œ šµ) āŠ† ran inr
3 djurf1o 9856 . . . . 5 inr:Vā€“1-1-ontoā†’({1o} Ɨ V)
4 f1of 6789 . . . . 5 (inr:Vā€“1-1-ontoā†’({1o} Ɨ V) ā†’ inr:VāŸ¶({1o} Ɨ V))
5 frn 6680 . . . . 5 (inr:VāŸ¶({1o} Ɨ V) ā†’ ran inr āŠ† ({1o} Ɨ V))
63, 4, 5mp2b 10 . . . 4 ran inr āŠ† ({1o} Ɨ V)
72, 6sstri 3958 . . 3 (inr ā€œ šµ) āŠ† ({1o} Ɨ V)
8 incom 4166 . . . 4 ((inl ā€œ š“) āˆ© ({1o} Ɨ V)) = (({1o} Ɨ V) āˆ© (inl ā€œ š“))
9 imassrn 6029 . . . . . 6 (inl ā€œ š“) āŠ† ran inl
10 djulf1o 9855 . . . . . . 7 inl:Vā€“1-1-ontoā†’({āˆ…} Ɨ V)
11 f1of 6789 . . . . . . 7 (inl:Vā€“1-1-ontoā†’({āˆ…} Ɨ V) ā†’ inl:VāŸ¶({āˆ…} Ɨ V))
12 frn 6680 . . . . . . 7 (inl:VāŸ¶({āˆ…} Ɨ V) ā†’ ran inl āŠ† ({āˆ…} Ɨ V))
1310, 11, 12mp2b 10 . . . . . 6 ran inl āŠ† ({āˆ…} Ɨ V)
149, 13sstri 3958 . . . . 5 (inl ā€œ š“) āŠ† ({āˆ…} Ɨ V)
15 1n0 8439 . . . . . . 7 1o ā‰  āˆ…
1615necomi 2999 . . . . . 6 āˆ… ā‰  1o
17 disjsn2 4678 . . . . . 6 (āˆ… ā‰  1o ā†’ ({āˆ…} āˆ© {1o}) = āˆ…)
18 xpdisj1 6118 . . . . . 6 (({āˆ…} āˆ© {1o}) = āˆ… ā†’ (({āˆ…} Ɨ V) āˆ© ({1o} Ɨ V)) = āˆ…)
1916, 17, 18mp2b 10 . . . . 5 (({āˆ…} Ɨ V) āˆ© ({1o} Ɨ V)) = āˆ…
20 ssdisj 4424 . . . . 5 (((inl ā€œ š“) āŠ† ({āˆ…} Ɨ V) āˆ§ (({āˆ…} Ɨ V) āˆ© ({1o} Ɨ V)) = āˆ…) ā†’ ((inl ā€œ š“) āˆ© ({1o} Ɨ V)) = āˆ…)
2114, 19, 20mp2an 691 . . . 4 ((inl ā€œ š“) āˆ© ({1o} Ɨ V)) = āˆ…
228, 21eqtr3i 2767 . . 3 (({1o} Ɨ V) āˆ© (inl ā€œ š“)) = āˆ…
23 ssdisj 4424 . . 3 (((inr ā€œ šµ) āŠ† ({1o} Ɨ V) āˆ§ (({1o} Ɨ V) āˆ© (inl ā€œ š“)) = āˆ…) ā†’ ((inr ā€œ šµ) āˆ© (inl ā€œ š“)) = āˆ…)
247, 22, 23mp2an 691 . 2 ((inr ā€œ šµ) āˆ© (inl ā€œ š“)) = āˆ…
251, 24eqtr3i 2767 1 ((inl ā€œ š“) āˆ© (inr ā€œ šµ)) = āˆ…
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ā‰  wne 2944  Vcvv 3448   āˆ© cin 3914   āŠ† wss 3915  āˆ…c0 4287  {csn 4591   Ɨ cxp 5636  ran crn 5639   ā€œ cima 5641  āŸ¶wf 6497  ā€“1-1-ontoā†’wf1o 6500  1oc1o 8410  inlcinl 9842  inrcinr 9843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1st 7926  df-2nd 7927  df-1o 8417  df-inl 9845  df-inr 9846
This theorem is referenced by: (None)
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