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Mirrors > Home > ILE Home > Th. List > axpre-apti | GIF version |
Description: Apartness of reals is
tight. Axiom for real and complex numbers,
derived from set theory. This construction-dependent theorem should not
be referenced directly; instead, use ax-pre-apti 7759.
(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-apti | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7660 | . . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7660 | . . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq1 3940 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
4 | breq2 3941 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 〈𝑦, 0R〉 <ℝ 𝐴)) | |
5 | 3, 4 | orbi12d 783 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
6 | 5 | notbid 657 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
7 | eqeq1 2147 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝐴 = 〈𝑦, 0R〉)) | |
8 | 6, 7 | imbi12d 233 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) ↔ (¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) → 𝐴 = 〈𝑦, 0R〉))) |
9 | breq2 3941 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
10 | breq1 3940 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
11 | 9, 10 | orbi12d 783 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
12 | 11 | notbid 657 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
13 | eqeq2 2150 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 = 〈𝑦, 0R〉 ↔ 𝐴 = 𝐵)) | |
14 | 12, 13 | imbi12d 233 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) → 𝐴 = 〈𝑦, 0R〉) ↔ (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵))) |
15 | aptisr 7611 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) → 𝑥 = 𝑦) | |
16 | 15 | 3expia 1184 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥) → 𝑥 = 𝑦)) |
17 | ltresr 7671 | . . . . . 6 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
18 | ltresr 7671 | . . . . . 6 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑦 <R 𝑥) | |
19 | 17, 18 | orbi12i 754 | . . . . 5 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) |
20 | 19 | notbii 658 | . . . 4 ⊢ (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) |
21 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
22 | 21 | eqresr 7668 | . . . 4 ⊢ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝑥 = 𝑦) |
23 | 16, 20, 22 | 3imtr4g 204 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉)) |
24 | 1, 2, 8, 14, 23 | 2gencl 2722 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵)) |
25 | 24 | 3impia 1179 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 〈cop 3535 class class class wbr 3937 Rcnr 7129 0Rc0r 7130 <R cltr 7135 ℝcr 7643 <ℝ cltrr 7648 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-i1p 7299 df-iplp 7300 df-iltp 7302 df-enr 7558 df-nr 7559 df-ltr 7562 df-0r 7563 df-r 7654 df-lt 7657 |
This theorem is referenced by: (None) |
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