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Mirrors > Home > ILE Home > Th. List > djuinr | Unicode version |
Description: The ranges of any left and right injections are disjoint. Remark: the extra generality offered by the two restrictions makes the theorem more readily usable (e.g., by djudom 6971 and djufun 6982) while the simpler statement inl inr is easily recovered from it by substituting for both and as done in casefun 6963). (Contributed by BJ and Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djuinr | inl inr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djulf1or 6934 | . . . 4 inl | |
2 | dff1o5 5369 | . . . . 5 inl inl inl | |
3 | 2 | simprbi 273 | . . . 4 inl inl |
4 | 1, 3 | ax-mp 5 | . . 3 inl |
5 | djurf1or 6935 | . . . 4 inr | |
6 | dff1o5 5369 | . . . . 5 inr inr inr | |
7 | 6 | simprbi 273 | . . . 4 inr inr |
8 | 5, 7 | ax-mp 5 | . . 3 inr |
9 | 4, 8 | ineq12i 3270 | . 2 inl inr |
10 | 1n0 6322 | . . . . 5 | |
11 | 10 | necomi 2391 | . . . 4 |
12 | disjsn2 3581 | . . . 4 | |
13 | 11, 12 | ax-mp 5 | . . 3 |
14 | xpdisj1 4958 | . . 3 | |
15 | 13, 14 | ax-mp 5 | . 2 |
16 | 9, 15 | eqtri 2158 | 1 inl inr |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wne 2306 cin 3065 c0 3358 csn 3522 cxp 4532 crn 4535 cres 4536 wf1 5115 wf1o 5117 c1o 6299 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-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: djuin 6942 casefun 6963 djudom 6971 djufun 6982 |
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