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Mirrors > Home > ILE Home > Th. List > exmidfodomrlemeldju | GIF version |
Description: Lemma for exmidfodomr 7234. A variant of djur 7099. (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 3170 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o) |
3 | el1o 6463 | . . . . . . . . 9 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
4 | 2, 3 | sylib 122 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅) |
5 | 4 | fveq2d 5538 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (inl‘𝑥) = (inl‘∅)) |
6 | 5 | eqeq2d 2201 | . . . . . 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 2607 | . . . 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 734 | . 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 5538 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅)) |
14 | 13 | eqeq2d 2201 | . . . . . 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 2607 | . . . 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 735 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
19 | exmidfodomrlemeldju.el | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) | |
20 | djur 7099 | . . 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 799 | 1 ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 ⊆ wss 3144 ∅c0 3437 ‘cfv 5235 1oc1o 6435 ⊔ cdju 7067 inlcinl 7075 inrcinr 7076 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-1st 6166 df-2nd 6167 df-1o 6442 df-dju 7068 df-inl 7077 df-inr 7078 |
This theorem is referenced by: exmidfodomrlemr 7232 |
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