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Mirrors > Home > ILE Home > Th. List > dju1p1e2 | GIF version |
Description: Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.) |
Ref | Expression |
---|---|
dju1p1e2 | ⊢ (1o ⊔ 1o) ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djuun 6952 | . 2 ⊢ ((inl “ 1o) ∪ (inr “ 1o)) = (1o ⊔ 1o) | |
2 | djuin 6949 | . . 3 ⊢ ((inl “ 1o) ∩ (inr “ 1o)) = ∅ | |
3 | djulf1o 6943 | . . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V) | |
4 | f1of1 5366 | . . . . . . . 8 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V)) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ inl:V–1-1→({∅} × V) |
6 | ssv 3119 | . . . . . . 7 ⊢ 1o ⊆ V | |
7 | f1ores 5382 | . . . . . . 7 ⊢ ((inl:V–1-1→({∅} × V) ∧ 1o ⊆ V) → (inl ↾ 1o):1o–1-1-onto→(inl “ 1o)) | |
8 | 5, 6, 7 | mp2an 422 | . . . . . 6 ⊢ (inl ↾ 1o):1o–1-1-onto→(inl “ 1o) |
9 | 1oex 6321 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | f1oen 6653 | . . . . . 6 ⊢ ((inl ↾ 1o):1o–1-1-onto→(inl “ 1o) → 1o ≈ (inl “ 1o)) |
11 | 8, 10 | ax-mp 5 | . . . . 5 ⊢ 1o ≈ (inl “ 1o) |
12 | 11 | ensymi 6676 | . . . 4 ⊢ (inl “ 1o) ≈ 1o |
13 | djurf1o 6944 | . . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V) | |
14 | f1of1 5366 | . . . . . . . 8 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V)) | |
15 | 13, 14 | ax-mp 5 | . . . . . . 7 ⊢ inr:V–1-1→({1o} × V) |
16 | f1ores 5382 | . . . . . . 7 ⊢ ((inr:V–1-1→({1o} × V) ∧ 1o ⊆ V) → (inr ↾ 1o):1o–1-1-onto→(inr “ 1o)) | |
17 | 15, 6, 16 | mp2an 422 | . . . . . 6 ⊢ (inr ↾ 1o):1o–1-1-onto→(inr “ 1o) |
18 | 9 | f1oen 6653 | . . . . . 6 ⊢ ((inr ↾ 1o):1o–1-1-onto→(inr “ 1o) → 1o ≈ (inr “ 1o)) |
19 | 17, 18 | ax-mp 5 | . . . . 5 ⊢ 1o ≈ (inr “ 1o) |
20 | 19 | ensymi 6676 | . . . 4 ⊢ (inr “ 1o) ≈ 1o |
21 | pm54.43 7046 | . . . 4 ⊢ (((inl “ 1o) ≈ 1o ∧ (inr “ 1o) ≈ 1o) → (((inl “ 1o) ∩ (inr “ 1o)) = ∅ ↔ ((inl “ 1o) ∪ (inr “ 1o)) ≈ 2o)) | |
22 | 12, 20, 21 | mp2an 422 | . . 3 ⊢ (((inl “ 1o) ∩ (inr “ 1o)) = ∅ ↔ ((inl “ 1o) ∪ (inr “ 1o)) ≈ 2o) |
23 | 2, 22 | mpbi 144 | . 2 ⊢ ((inl “ 1o) ∪ (inr “ 1o)) ≈ 2o |
24 | 1, 23 | eqbrtrri 3951 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 Vcvv 2686 ∪ cun 3069 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 {csn 3527 class class class wbr 3929 × cxp 4537 ↾ cres 4541 “ cima 4542 –1-1→wf1 5120 –1-1-onto→wf1o 5122 1oc1o 6306 2oc2o 6307 ≈ cen 6632 ⊔ cdju 6922 inlcinl 6930 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 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 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-int 3772 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-iom 4505 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-2o 6314 df-er 6429 df-en 6635 df-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: exmidfodomrlemr 7058 exmidfodomrlemrALT 7059 |
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