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 7860.
(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 7761 | . . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7761 | . . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq1 3980 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
4 | breq2 3981 | . . . . . 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 2171 | . . . 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 3981 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
10 | breq1 3980 | . . . . . 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 2174 | . . . 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 7712 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) → 𝑥 = 𝑦) | |
16 | 15 | 3expia 1194 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥) → 𝑥 = 𝑦)) |
17 | ltresr 7772 | . . . . . 6 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
18 | ltresr 7772 | . . . . . 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 2725 | . . . . 5 ⊢ 𝑥 ∈ V | |
22 | 21 | eqresr 7769 | . . . 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 2755 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵)) |
25 | 24 | 3impia 1189 | 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 967 = wceq 1342 ∈ wcel 2135 〈cop 3574 class class class wbr 3977 Rcnr 7230 0Rc0r 7231 <R cltr 7236 ℝcr 7744 <ℝ cltrr 7749 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-eprel 4262 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-1o 6376 df-2o 6377 df-oadd 6380 df-omul 6381 df-er 6493 df-ec 6495 df-qs 6499 df-ni 7237 df-pli 7238 df-mi 7239 df-lti 7240 df-plpq 7277 df-mpq 7278 df-enq 7280 df-nqqs 7281 df-plqqs 7282 df-mqqs 7283 df-1nqqs 7284 df-rq 7285 df-ltnqqs 7286 df-enq0 7357 df-nq0 7358 df-0nq0 7359 df-plq0 7360 df-mq0 7361 df-inp 7399 df-i1p 7400 df-iplp 7401 df-iltp 7403 df-enr 7659 df-nr 7660 df-ltr 7663 df-0r 7664 df-r 7755 df-lt 7758 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |