djuin - Intuitionistic Logic Explorer

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Theorem djuin 7037

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 4622	. . 3 inl inl
2		df-ima 4622	. . 3 inr inr
3	1, 2	ineq12i 3326	. 2 inl inr inl inr
4		djuinr 7036	. 2 inl inr
5	3, 4	eqtri 2191	1 inl inr

Colors of variables: wff set class

Syntax hints:

wceq 1348

cin 3120

c0 3414

crn 4610

cres 4611

cima 4612 inlcinl 7018 inrcinr 7019

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-1st 6116 df-2nd 6117 df-1o 6392 df-inl 7020 df-inr 7021

This theorem is referenced by: caseinl 7064 caseinr 7065 endjusym 7069 ctssdccl 7084 dju1p1e2 7161 endjudisj 7174 djuen 7175

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