![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > exmidfodomrlemeldju | GIF version |
Description: Lemma for exmidfodomr 6891. A variant of djur 6811. (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 3026 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o) |
3 | el1o 6215 | . . . . . . . . 9 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅) | |
4 | 2, 3 | sylib 121 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅) |
5 | 4 | fveq2d 5322 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (inl‘𝑥) = (inl‘∅)) |
6 | 5 | eqeq2d 2100 | . . . . . 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 2490 | . . . 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 688 | . 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 5322 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅)) |
14 | 13 | eqeq2d 2100 | . . . . . 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 2490 | . . . 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 689 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
19 | exmidfodomrlemeldju.el | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o)) | |
20 | djur 6811 | . . 3 ⊢ (𝐵 ∈ (𝐴 ⊔ 1o) → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥))) | |
21 | 19, 20 | syl 14 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥))) |
22 | 10, 18, 21 | mpjaodan 748 | 1 ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 665 = wceq 1290 ∈ wcel 1439 ∃wrex 2361 ⊆ wss 3000 ∅c0 3287 ‘cfv 5028 1oc1o 6188 ⊔ cdju 6784 inlcinl 6791 inrcinr 6792 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-fo 5034 df-fv 5036 df-1st 5925 df-2nd 5926 df-1o 6195 df-dju 6785 df-inl 6793 df-inr 6794 |
This theorem is referenced by: exmidfodomrlemr 6889 |
Copyright terms: Public domain | W3C validator |