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Mirrors > Home > ILE Home > Th. List > exmidfodomrlemeldju | GIF version |
Description: Lemma for exmidfodomr 7005. A variant of djur 6904. (Contributed by Jim Kingdon, 2-Jul-2022.) |
Ref | Expression |
---|---|
exmidfodomrlemeldju.a | ⊢ (𝜑 → 𝐴 ⊆ 1o) |
exmidfodomrlemeldju.el | ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) |
Ref | Expression |
---|---|
exmidfodomrlemeldju | ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidfodomrlemeldju.a | . . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 1o) | |
2 | 1 | sselda 3061 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o) |
3 | el1o 6286 | . . . . . . . . 9 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅) |
5 | 4 | fveq2d 5377 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (inl‘𝑥) = (inl‘∅)) |
6 | 5 | eqeq2d 2124 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅))) |
7 | 6 | biimpd 143 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅))) |
8 | 7 | rexlimdva 2521 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅))) |
9 | 8 | imp 123 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅)) |
10 | 9 | orcd 705 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
11 | simpr 109 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 ∈ 1o) | |
12 | 11, 3 | sylib 121 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 = ∅) |
13 | 12 | fveq2d 5377 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅)) |
14 | 13 | eqeq2d 2124 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅))) |
15 | 14 | biimpd 143 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅))) |
16 | 15 | rexlimdva 2521 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅))) |
17 | 16 | imp 123 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅)) |
18 | 17 | olcd 706 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
19 | exmidfodomrlemeldju.el | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) | |
20 | djur 6904 | . . 3 ⊢ (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥))) | |
21 | 19, 20 | sylib 121 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥))) |
22 | 10, 18, 21 | mpjaodan 770 | 1 ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 680 = wceq 1312 ∈ wcel 1461 ∃wrex 2389 ⊆ wss 3035 ∅c0 3327 ‘cfv 5079 1oc1o 6258 ⊔ cdju 6872 inlcinl 6880 inrcinr 6881 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-iord 4246 df-on 4248 df-suc 4251 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-1st 5990 df-2nd 5991 df-1o 6265 df-dju 6873 df-inl 6882 df-inr 6883 |
This theorem is referenced by: exmidfodomrlemr 7003 |
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