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Mirrors > Home > ILE Home > Th. List > exmidfodomrlemeldju | GIF version |
Description: Lemma for exmidfodomr 7196. A variant of djur 7061. (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 3155 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o) |
3 | el1o 6431 | . . . . . . . . 9 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
4 | 2, 3 | sylib 122 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅) |
5 | 4 | fveq2d 5514 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (inl‘𝑥) = (inl‘∅)) |
6 | 5 | eqeq2d 2189 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅))) |
7 | 6 | biimpd 144 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅))) |
8 | 7 | rexlimdva 2594 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅))) |
9 | 8 | imp 124 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅)) |
10 | 9 | orcd 733 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
11 | simpr 110 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 ∈ 1o) | |
12 | 11, 3 | sylib 122 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 = ∅) |
13 | 12 | fveq2d 5514 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅)) |
14 | 13 | eqeq2d 2189 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅))) |
15 | 14 | biimpd 144 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅))) |
16 | 15 | rexlimdva 2594 | . . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅))) |
17 | 16 | imp 124 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅)) |
18 | 17 | olcd 734 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
19 | exmidfodomrlemeldju.el | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) | |
20 | djur 7061 | . . 3 ⊢ (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥))) | |
21 | 19, 20 | sylib 122 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥))) |
22 | 10, 18, 21 | mpjaodan 798 | 1 ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 ⊆ wss 3129 ∅c0 3422 ‘cfv 5211 1oc1o 6403 ⊔ 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-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 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-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 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-dju 7030 df-inl 7039 df-inr 7040 |
This theorem is referenced by: exmidfodomrlemr 7194 |
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