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Mirrors > Home > ILE Home > Th. List > dju1p1e2 | Unicode version |
Description: Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
Ref | Expression |
---|---|
dju1p1e2 | ⊔ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuun 6945 | . 2 inl inr ⊔ | |
2 | djuin 6942 | . . 3 inl inr | |
3 | djulf1o 6936 | . . . . . . . 8 inl | |
4 | f1of1 5359 | . . . . . . . 8 inl inl | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 inl |
6 | ssv 3114 | . . . . . . 7 | |
7 | f1ores 5375 | . . . . . . 7 inl inl inl | |
8 | 5, 6, 7 | mp2an 422 | . . . . . 6 inl inl |
9 | 1oex 6314 | . . . . . . 7 | |
10 | 9 | f1oen 6646 | . . . . . 6 inl inl inl |
11 | 8, 10 | ax-mp 5 | . . . . 5 inl |
12 | 11 | ensymi 6669 | . . . 4 inl |
13 | djurf1o 6937 | . . . . . . . 8 inr | |
14 | f1of1 5359 | . . . . . . . 8 inr inr | |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 inr |
16 | f1ores 5375 | . . . . . . 7 inr inr inr | |
17 | 15, 6, 16 | mp2an 422 | . . . . . 6 inr inr |
18 | 9 | f1oen 6646 | . . . . . 6 inr inr inr |
19 | 17, 18 | ax-mp 5 | . . . . 5 inr |
20 | 19 | ensymi 6669 | . . . 4 inr |
21 | pm54.43 7039 | . . . 4 inl inr inl inr inl inr | |
22 | 12, 20, 21 | mp2an 422 | . . 3 inl inr inl inr |
23 | 2, 22 | mpbi 144 | . 2 inl inr |
24 | 1, 23 | eqbrtrri 3946 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1331 cvv 2681 cun 3064 cin 3065 wss 3066 c0 3358 csn 3522 class class class wbr 3924 cxp 4532 cres 4536 cima 4537 wf1 5115 wf1o 5117 c1o 6299 c2o 6300 cen 6625 ⊔ cdju 6915 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-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 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-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-int 3767 df-iun 3810 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-iom 4500 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-2o 6307 df-er 6422 df-en 6628 df-dju 6916 df-inl 6925 df-inr 6926 |
This theorem is referenced by: exmidfodomrlemr 7051 exmidfodomrlemrALT 7052 |
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