Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7091.
(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 6997 | . . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 6997 | . . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq1 3788 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
4 | breq2 3789 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 〈𝑦, 0R〉 <ℝ 𝐴)) | |
5 | 3, 4 | orbi12d 739 | . . . . 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 624 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
7 | eqeq1 2087 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝐴 = 〈𝑦, 0R〉)) | |
8 | 6, 7 | imbi12d 232 | . . 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 3789 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
10 | breq1 3788 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
11 | 9, 10 | orbi12d 739 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
12 | 11 | notbid 624 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
13 | eqeq2 2090 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 = 〈𝑦, 0R〉 ↔ 𝐴 = 𝐵)) | |
14 | 12, 13 | imbi12d 232 | . . 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 6955 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) → 𝑥 = 𝑦) | |
16 | 15 | 3expia 1140 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥) → 𝑥 = 𝑦)) |
17 | ltresr 7007 | . . . . . 6 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
18 | ltresr 7007 | . . . . . 6 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑦 <R 𝑥) | |
19 | 17, 18 | orbi12i 713 | . . . . 5 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) |
20 | 19 | notbii 626 | . . . 4 ⊢ (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) |
21 | vex 2604 | . . . . 5 ⊢ 𝑥 ∈ V | |
22 | 21 | eqresr 7004 | . . . 4 ⊢ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝑥 = 𝑦) |
23 | 16, 20, 22 | 3imtr4g 203 | . . 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 2632 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵)) |
25 | 24 | 3impia 1135 | 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 102 ∨ wo 661 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 〈cop 3401 class class class wbr 3785 Rcnr 6487 0Rc0r 6488 <R cltr 6493 ℝcr 6980 <ℝ cltrr 6985 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-iltp 6660 df-enr 6903 df-nr 6904 df-ltr 6907 df-0r 6908 df-r 6991 df-lt 6994 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |