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Mirrors > Home > ILE Home > Th. List > dedekindicclemeu | GIF version |
Description: Lemma for dedekindicc 12780. Part of proving uniqueness. (Contributed by Jim Kingdon, 15-Feb-2024.) |
Ref | Expression |
---|---|
dedekindicc.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dedekindicc.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
dedekindicc.lss | ⊢ (𝜑 → 𝐿 ⊆ (𝐴[,]𝐵)) |
dedekindicc.uss | ⊢ (𝜑 → 𝑈 ⊆ (𝐴[,]𝐵)) |
dedekindicc.lm | ⊢ (𝜑 → ∃𝑞 ∈ (𝐴[,]𝐵)𝑞 ∈ 𝐿) |
dedekindicc.um | ⊢ (𝜑 → ∃𝑟 ∈ (𝐴[,]𝐵)𝑟 ∈ 𝑈) |
dedekindicc.lr | ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)(𝑞 ∈ 𝐿 ↔ ∃𝑟 ∈ 𝐿 𝑞 < 𝑟)) |
dedekindicc.ur | ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝑟 ∈ 𝑈 ↔ ∃𝑞 ∈ 𝑈 𝑞 < 𝑟)) |
dedekindicc.disj | ⊢ (𝜑 → (𝐿 ∩ 𝑈) = ∅) |
dedekindicc.loc | ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
dedekindicc.ab | ⊢ (𝜑 → 𝐴 < 𝐵) |
dedekindicclemeu.are | ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) |
dedekindicclemeu.ac | ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟)) |
dedekindicclemeu.bre | ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) |
dedekindicclemeu.bc | ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟)) |
dedekindicclemeu.lt | ⊢ (𝜑 → 𝐶 < 𝐷) |
Ref | Expression |
---|---|
dedekindicclemeu | ⊢ (𝜑 → ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3932 | . . . 4 ⊢ (𝑞 = 𝐶 → (𝑞 < 𝐶 ↔ 𝐶 < 𝐶)) | |
2 | dedekindicclemeu.ac | . . . . . 6 ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐶 ∧ ∀𝑟 ∈ 𝑈 𝐶 < 𝑟)) | |
3 | 2 | simpld 111 | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ 𝐿 𝑞 < 𝐶) |
4 | 3 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → ∀𝑞 ∈ 𝐿 𝑞 < 𝐶) |
5 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → 𝐶 ∈ 𝐿) | |
6 | 1, 4, 5 | rspcdva 2794 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → 𝐶 < 𝐶) |
7 | dedekindicc.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
8 | dedekindicc.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | iccssre 9738 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
10 | 7, 8, 9 | syl2anc 408 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
11 | dedekindicclemeu.are | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵)) | |
12 | 10, 11 | sseldd 3098 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | 12 | ltnrd 7875 | . . . 4 ⊢ (𝜑 → ¬ 𝐶 < 𝐶) |
14 | 13 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → ¬ 𝐶 < 𝐶) |
15 | 6, 14 | pm2.21fal 1351 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐿) → ⊥) |
16 | breq2 3933 | . . . 4 ⊢ (𝑟 = 𝐷 → (𝐷 < 𝑟 ↔ 𝐷 < 𝐷)) | |
17 | dedekindicclemeu.bc | . . . . . 6 ⊢ (𝜑 → (∀𝑞 ∈ 𝐿 𝑞 < 𝐷 ∧ ∀𝑟 ∈ 𝑈 𝐷 < 𝑟)) | |
18 | 17 | simprd 113 | . . . . 5 ⊢ (𝜑 → ∀𝑟 ∈ 𝑈 𝐷 < 𝑟) |
19 | 18 | adantr 274 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → ∀𝑟 ∈ 𝑈 𝐷 < 𝑟) |
20 | simpr 109 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → 𝐷 ∈ 𝑈) | |
21 | 16, 19, 20 | rspcdva 2794 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → 𝐷 < 𝐷) |
22 | dedekindicclemeu.bre | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (𝐴[,]𝐵)) | |
23 | 10, 22 | sseldd 3098 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
24 | 23 | ltnrd 7875 | . . . 4 ⊢ (𝜑 → ¬ 𝐷 < 𝐷) |
25 | 24 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → ¬ 𝐷 < 𝐷) |
26 | 21, 25 | pm2.21fal 1351 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝑈) → ⊥) |
27 | dedekindicclemeu.lt | . . 3 ⊢ (𝜑 → 𝐶 < 𝐷) | |
28 | breq2 3933 | . . . . 5 ⊢ (𝑟 = 𝐷 → (𝐶 < 𝑟 ↔ 𝐶 < 𝐷)) | |
29 | eleq1 2202 | . . . . . 6 ⊢ (𝑟 = 𝐷 → (𝑟 ∈ 𝑈 ↔ 𝐷 ∈ 𝑈)) | |
30 | 29 | orbi2d 779 | . . . . 5 ⊢ (𝑟 = 𝐷 → ((𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈) ↔ (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈))) |
31 | 28, 30 | imbi12d 233 | . . . 4 ⊢ (𝑟 = 𝐷 → ((𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)) ↔ (𝐶 < 𝐷 → (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈)))) |
32 | breq1 3932 | . . . . . . 7 ⊢ (𝑞 = 𝐶 → (𝑞 < 𝑟 ↔ 𝐶 < 𝑟)) | |
33 | eleq1 2202 | . . . . . . . 8 ⊢ (𝑞 = 𝐶 → (𝑞 ∈ 𝐿 ↔ 𝐶 ∈ 𝐿)) | |
34 | 33 | orbi1d 780 | . . . . . . 7 ⊢ (𝑞 = 𝐶 → ((𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈) ↔ (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
35 | 32, 34 | imbi12d 233 | . . . . . 6 ⊢ (𝑞 = 𝐶 → ((𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)) ↔ (𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))) |
36 | 35 | ralbidv 2437 | . . . . 5 ⊢ (𝑞 = 𝐶 → (∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)) ↔ ∀𝑟 ∈ (𝐴[,]𝐵)(𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈)))) |
37 | dedekindicc.loc | . . . . 5 ⊢ (𝜑 → ∀𝑞 ∈ (𝐴[,]𝐵)∀𝑟 ∈ (𝐴[,]𝐵)(𝑞 < 𝑟 → (𝑞 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) | |
38 | 36, 37, 11 | rspcdva 2794 | . . . 4 ⊢ (𝜑 → ∀𝑟 ∈ (𝐴[,]𝐵)(𝐶 < 𝑟 → (𝐶 ∈ 𝐿 ∨ 𝑟 ∈ 𝑈))) |
39 | 31, 38, 22 | rspcdva 2794 | . . 3 ⊢ (𝜑 → (𝐶 < 𝐷 → (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈))) |
40 | 27, 39 | mpd 13 | . 2 ⊢ (𝜑 → (𝐶 ∈ 𝐿 ∨ 𝐷 ∈ 𝑈)) |
41 | 15, 26, 40 | mpjaodan 787 | 1 ⊢ (𝜑 → ⊥) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 697 = wceq 1331 ⊥wfal 1336 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 ∩ cin 3070 ⊆ wss 3071 ∅c0 3363 class class class wbr 3929 (class class class)co 5774 ℝcr 7619 < clt 7800 [,]cicc 9674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-icc 9678 |
This theorem is referenced by: dedekindicclemicc 12779 |
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