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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 7059 | . 2 ⊢ ((inl “ 1o) ∪ (inr “ 1o)) = (1o ⊔ 1o) | |
2 | djuin 7056 | . . 3 ⊢ ((inl “ 1o) ∩ (inr “ 1o)) = ∅ | |
3 | djulf1o 7050 | . . . . . . . 8 ⊢ inl:V–1-1-onto→({∅} × V) | |
4 | f1of1 5455 | . . . . . . . 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 3177 | . . . . . . 7 ⊢ 1o ⊆ V | |
7 | f1ores 5471 | . . . . . . 7 ⊢ ((inl:V–1-1→({∅} × V) ∧ 1o ⊆ V) → (inl ↾ 1o):1o–1-1-onto→(inl “ 1o)) | |
8 | 5, 6, 7 | mp2an 426 | . . . . . 6 ⊢ (inl ↾ 1o):1o–1-1-onto→(inl “ 1o) |
9 | 1oex 6418 | . . . . . . 7 ⊢ 1o ∈ V | |
10 | 9 | f1oen 6752 | . . . . . 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 6775 | . . . 4 ⊢ (inl “ 1o) ≈ 1o |
13 | djurf1o 7051 | . . . . . . . 8 ⊢ inr:V–1-1-onto→({1o} × V) | |
14 | f1of1 5455 | . . . . . . . 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 5471 | . . . . . . 7 ⊢ ((inr:V–1-1→({1o} × V) ∧ 1o ⊆ V) → (inr ↾ 1o):1o–1-1-onto→(inr “ 1o)) | |
17 | 15, 6, 16 | mp2an 426 | . . . . . 6 ⊢ (inr ↾ 1o):1o–1-1-onto→(inr “ 1o) |
18 | 9 | f1oen 6752 | . . . . . 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 6775 | . . . 4 ⊢ (inr “ 1o) ≈ 1o |
21 | pm54.43 7182 | . . . 4 ⊢ (((inl “ 1o) ≈ 1o ∧ (inr “ 1o) ≈ 1o) → (((inl “ 1o) ∩ (inr “ 1o)) = ∅ ↔ ((inl “ 1o) ∪ (inr “ 1o)) ≈ 2o)) | |
22 | 12, 20, 21 | mp2an 426 | . . 3 ⊢ (((inl “ 1o) ∩ (inr “ 1o)) = ∅ ↔ ((inl “ 1o) ∪ (inr “ 1o)) ≈ 2o) |
23 | 2, 22 | mpbi 145 | . 2 ⊢ ((inl “ 1o) ∪ (inr “ 1o)) ≈ 2o |
24 | 1, 23 | eqbrtrri 4023 | 1 ⊢ (1o ⊔ 1o) ≈ 2o |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 Vcvv 2737 ∪ cun 3127 ∩ cin 3128 ⊆ wss 3129 ∅c0 3422 {csn 3591 class class class wbr 4000 × cxp 4620 ↾ cres 4624 “ cima 4625 –1-1→wf1 5208 –1-1-onto→wf1o 5210 1oc1o 6403 2oc2o 6404 ≈ cen 6731 ⊔ cdju 7029 inlcinl 7037 inrcinr 7038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-1st 6134 df-2nd 6135 df-1o 6410 df-2o 6411 df-er 6528 df-en 6734 df-dju 7030 df-inl 7039 df-inr 7040 |
This theorem is referenced by: exmidfodomrlemr 7194 exmidfodomrlemrALT 7195 |
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