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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 6920 | . 2 ⊢ ((inl “ 1o) ∪ (inr “ 1o)) = (1o ⊔ 1o) | |
2 | djuin 6917 | . . 3 ⊢ ((inl “ 1o) ∩ (inr “ 1o)) = ∅ | |
3 | djulf1o 6911 | . . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V) | |
4 | f1of1 5334 | . . . . . . . 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 3089 | . . . . . . 7 ⊢ 1o ⊆ V | |
7 | f1ores 5350 | . . . . . . 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 6289 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | f1oen 6621 | . . . . . 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 6644 | . . . 4 ⊢ (inl “ 1o) ≈ 1o |
13 | djurf1o 6912 | . . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V) | |
14 | f1of1 5334 | . . . . . . . 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 5350 | . . . . . . 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 6621 | . . . . . 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 6644 | . . . 4 ⊢ (inr “ 1o) ≈ 1o |
21 | pm54.43 7014 | . . . 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 3921 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 Vcvv 2660 ∪ cun 3039 ∩ cin 3040 ⊆ wss 3041 ∅c0 3333 {csn 3497 class class class wbr 3899 × cxp 4507 ↾ cres 4511 “ cima 4512 –1-1→wf1 5090 –1-1-onto→wf1o 5092 1oc1o 6274 2oc2o 6275 ≈ cen 6600 ⊔ cdju 6890 inlcinl 6898 inrcinr 6899 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1st 6006 df-2nd 6007 df-1o 6281 df-2o 6282 df-er 6397 df-en 6603 df-dju 6891 df-inl 6900 df-inr 6901 |
This theorem is referenced by: exmidfodomrlemr 7026 exmidfodomrlemrALT 7027 |
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