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Mirrors > Home > ILE Home > Th. List > reapcotr | GIF version |
Description: Real apartness is cotransitive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 16-Feb-2020.) |
Ref | Expression |
---|---|
reapcotr | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reaplt 8575 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) | |
2 | 1 | 3adant3 1019 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
3 | axltwlin 8055 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐵 → (𝐴 < 𝐶 ∨ 𝐶 < 𝐵))) | |
4 | axltwlin 8055 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐴 → (𝐵 < 𝐶 ∨ 𝐶 < 𝐴))) | |
5 | 4 | 3com12 1209 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 < 𝐴 → (𝐵 < 𝐶 ∨ 𝐶 < 𝐴))) |
6 | 3, 5 | orim12d 787 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐴)))) |
7 | 2, 6 | sylbid 150 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → ((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐴)))) |
8 | orcom 729 | . . . . 5 ⊢ ((𝐵 < 𝐶 ∨ 𝐶 < 𝐴) ↔ (𝐶 < 𝐴 ∨ 𝐵 < 𝐶)) | |
9 | 8 | orbi2i 763 | . . . 4 ⊢ (((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐴)) ↔ ((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ∨ (𝐶 < 𝐴 ∨ 𝐵 < 𝐶))) |
10 | or42 773 | . . . 4 ⊢ (((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ∨ (𝐶 < 𝐴 ∨ 𝐵 < 𝐶)) ↔ ((𝐴 < 𝐶 ∨ 𝐶 < 𝐴) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐵))) | |
11 | 9, 10 | bitri 184 | . . 3 ⊢ (((𝐴 < 𝐶 ∨ 𝐶 < 𝐵) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐴)) ↔ ((𝐴 < 𝐶 ∨ 𝐶 < 𝐴) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐵))) |
12 | 7, 11 | imbitrdi 161 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → ((𝐴 < 𝐶 ∨ 𝐶 < 𝐴) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐵)))) |
13 | reaplt 8575 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐶 ↔ (𝐴 < 𝐶 ∨ 𝐶 < 𝐴))) | |
14 | 13 | 3adant2 1018 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐶 ↔ (𝐴 < 𝐶 ∨ 𝐶 < 𝐴))) |
15 | reaplt 8575 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 # 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐶 < 𝐵))) | |
16 | 15 | 3adant1 1017 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 # 𝐶 ↔ (𝐵 < 𝐶 ∨ 𝐶 < 𝐵))) |
17 | 14, 16 | orbi12d 794 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 # 𝐶 ∨ 𝐵 # 𝐶) ↔ ((𝐴 < 𝐶 ∨ 𝐶 < 𝐴) ∨ (𝐵 < 𝐶 ∨ 𝐶 < 𝐵)))) |
18 | 12, 17 | sylibrd 169 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 # 𝐵 → (𝐴 # 𝐶 ∨ 𝐵 # 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 709 ∧ w3a 980 ∈ wcel 2160 class class class wbr 4018 ℝcr 7840 < clt 8022 # cap 8568 |
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-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-1re 7935 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-mulrcl 7940 ax-addcom 7941 ax-mulcom 7942 ax-addass 7943 ax-mulass 7944 ax-distr 7945 ax-i2m1 7946 ax-0lt1 7947 ax-1rid 7948 ax-0id 7949 ax-rnegex 7950 ax-precex 7951 ax-cnre 7952 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 ax-pre-apti 7956 ax-pre-ltadd 7957 ax-pre-mulgt0 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 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-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-ltxr 8027 df-sub 8160 df-neg 8161 df-reap 8562 df-ap 8569 |
This theorem is referenced by: apcotr 8594 |
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