djuin - Intuitionistic Logic Explorer

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Theorem djuin 6942

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 inr

**Proof of Theorem djuin**
Step	Hyp	Ref	Expression
1		df-ima 4547	. . 3 inl inl
2		df-ima 4547	. . 3 inr inr
3	1, 2	ineq12i 3270	. 2 inl inr inl inr
4		djuinr 6941	. 2 inl inr
5	3, 4	eqtri 2158	1 inl inr

Colors of variables: wff set class

Syntax hints:

wceq 1331

cin 3065

c0 3358

crn 4535

cres 4536

cima 4537 inlcinl 6923 inrcinr 6924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1st 6031 df-2nd 6032 df-1o 6306 df-inl 6925 df-inr 6926

This theorem is referenced by: caseinl 6969 caseinr 6970 endjusym 6974 ctssdccl 6989 dju1p1e2 7046 endjudisj 7059 djuen 7060

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