Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7040 | . 2 inl inr ⊔ | |
2 | djuin 7037 | . . 3 inl inr | |
3 | djulf1o 7031 | . . . . . . . 8 inl | |
4 | f1of1 5439 | . . . . . . . 8 inl inl | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 inl |
6 | ssv 3169 | . . . . . . 7 | |
7 | f1ores 5455 | . . . . . . 7 inl inl inl | |
8 | 5, 6, 7 | mp2an 424 | . . . . . 6 inl inl |
9 | 1oex 6400 | . . . . . . 7 | |
10 | 9 | f1oen 6733 | . . . . . 6 inl inl inl |
11 | 8, 10 | ax-mp 5 | . . . . 5 inl |
12 | 11 | ensymi 6756 | . . . 4 inl |
13 | djurf1o 7032 | . . . . . . . 8 inr | |
14 | f1of1 5439 | . . . . . . . 8 inr inr | |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 inr |
16 | f1ores 5455 | . . . . . . 7 inr inr inr | |
17 | 15, 6, 16 | mp2an 424 | . . . . . 6 inr inr |
18 | 9 | f1oen 6733 | . . . . . 6 inr inr inr |
19 | 17, 18 | ax-mp 5 | . . . . 5 inr |
20 | 19 | ensymi 6756 | . . . 4 inr |
21 | pm54.43 7154 | . . . 4 inl inr inl inr inl inr | |
22 | 12, 20, 21 | mp2an 424 | . . 3 inl inr inl inr |
23 | 2, 22 | mpbi 144 | . 2 inl inr |
24 | 1, 23 | eqbrtrri 4010 | 1 ⊔ |
Colors of variables: wff set class |
Syntax hints: wb 104 wceq 1348 cvv 2730 cun 3119 cin 3120 wss 3121 c0 3414 csn 3581 class class class wbr 3987 cxp 4607 cres 4611 cima 4612 wf1 5193 wf1o 5195 c1o 6385 c2o 6386 cen 6712 ⊔ cdju 7010 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-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-int 3830 df-iun 3873 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-iom 4573 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-2o 6393 df-er 6509 df-en 6715 df-dju 7011 df-inl 7020 df-inr 7021 |
This theorem is referenced by: exmidfodomrlemr 7166 exmidfodomrlemrALT 7167 |
Copyright terms: Public domain | W3C validator |