Theorem djuin 7404

Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) (Proof shortened by BJ, 9-Jul-2023.)

Assertion

Ref	Expression
djuin	|- ( (inl " A ) i^i (inr " B ) ) = (/)

Proof of Theorem djuin

Step	Hyp	Ref	Expression
1		df-ima 4787	. . 3 |- (inl " A ) = ran (inl |` A )
2		df-ima 4787	. . 3 |- (inr " B ) = ran (inr |` B )
3	1, 2	ineq12i 3430	. 2 |- ( (inl " A ) i^i (inr " B ) ) = ( ran (inl |` A ) i^i ran (inr |` B ) )
4		djuinr 7403	. 2 |- ( ran (inl |` A ) i^i ran (inr |` B ) ) = (/)
5	3, 4	eqtri 2259	1 |- ( (inl " A ) i^i (inr " B ) ) = (/)

This proof depends on syntax axioms:   = wceq 1402   i^i cin 3219   (/)c0 3520   ran crn 4775   |` cres 4776   "cima 4777  inlcinl 7385  inrcinr 7386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1st 6374  df-2nd 6375  df-1o 6687  df-inl 7387  df-inr 7388

This theorem is used by:  caseinl  7431  caseinr  7432  endjusym  7436  ctssdccl  7451  dju1p1e2  7549  endjudisj  7566  djuen  7567