Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > djudomr | Unicode version |
Description: A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.) |
Ref | Expression |
---|---|
djudomr | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inr 6933 | . . . . 5 inr | |
2 | 1 | funmpt2 5162 | . . . 4 inr |
3 | simpr 109 | . . . 4 | |
4 | resfunexg 5641 | . . . 4 inr inr | |
5 | 2, 3, 4 | sylancr 410 | . . 3 inr |
6 | inrresf1 6947 | . . 3 inr ⊔ | |
7 | f1eq1 5323 | . . . 4 inr ⊔ inr ⊔ | |
8 | 7 | spcegv 2774 | . . 3 inr inr ⊔ ⊔ |
9 | 5, 6, 8 | mpisyl 1422 | . 2 ⊔ |
10 | djuex 6928 | . . 3 ⊔ | |
11 | brdomg 6642 | . . 3 ⊔ ⊔ ⊔ | |
12 | 10, 11 | syl 14 | . 2 ⊔ ⊔ |
13 | 9, 12 | mpbird 166 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wex 1468 wcel 1480 cvv 2686 cop 3530 class class class wbr 3929 cres 4541 wfun 5117 wf1 5120 c1o 6306 cdom 6633 ⊔ cdju 6922 inrcinr 6931 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dom 6636 df-dju 6923 df-inr 6933 |
This theorem is referenced by: sbthom 13221 |
Copyright terms: Public domain | W3C validator |