MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  djuin Structured version   Visualization version   GIF version

Theorem djuin 9941
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 4201 . 2 ((inr ā€œ šµ) ∩ (inl ā€œ š“)) = ((inl ā€œ š“) ∩ (inr ā€œ šµ))
2 imassrn 6074 . . . 4 (inr ā€œ šµ) āŠ† ran inr
3 djurf1o 9936 . . . . 5 inr:V–1-1-onto→({1o} Ɨ V)
4 f1of 6839 . . . . 5 (inr:V–1-1-onto→({1o} Ɨ V) → inr:V⟶({1o} Ɨ V))
5 frn 6729 . . . . 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 3989 . . 3 (inr ā€œ šµ) āŠ† ({1o} Ɨ V)
8 incom 4201 . . . 4 ((inl ā€œ š“) ∩ ({1o} Ɨ V)) = (({1o} Ɨ V) ∩ (inl ā€œ š“))
9 imassrn 6074 . . . . . 6 (inl ā€œ š“) āŠ† ran inl
10 djulf1o 9935 . . . . . . 7 inl:V–1-1-onto→({āˆ…} Ɨ V)
11 f1of 6839 . . . . . . 7 (inl:V–1-1-onto→({āˆ…} Ɨ V) → inl:V⟶({āˆ…} Ɨ V))
12 frn 6729 . . . . . . 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 3989 . . . . 5 (inl ā€œ š“) āŠ† ({āˆ…} Ɨ V)
15 1n0 8508 . . . . . . 7 1o ≠ āˆ…
1615necomi 2992 . . . . . 6 āˆ… ≠ 1o
17 disjsn2 4717 . . . . . 6 (āˆ… ≠ 1o → ({āˆ…} ∩ {1o}) = āˆ…)
18 xpdisj1 6165 . . . . . 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 4460 . . . . 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 691 . . . 4 ((inl ā€œ š“) ∩ ({1o} Ɨ V)) = āˆ…
228, 21eqtr3i 2758 . . 3 (({1o} Ɨ V) ∩ (inl ā€œ š“)) = āˆ…
23 ssdisj 4460 . . 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 691 . 2 ((inr ā€œ šµ) ∩ (inl ā€œ š“)) = āˆ…
251, 24eqtr3i 2758 1 ((inl ā€œ š“) ∩ (inr ā€œ šµ)) = āˆ…
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534   ≠ wne 2937  Vcvv 3471   ∩ cin 3946   āŠ† wss 3947  āˆ…c0 4323  {csn 4629   Ɨ cxp 5676  ran crn 5679   ā€œ cima 5681  āŸ¶wf 6544  ā€“1-1-onto→wf1o 6547  1oc1o 8479  inlcinl 9922  inrcinr 9923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-om 7871  df-1st 7993  df-2nd 7994  df-1o 8486  df-inl 9925  df-inr 9926
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator