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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 6798.
(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 6727 | . . 3 ⊢ (A ∈ ℝ ↔ ∃x ∈ R 〈x, 0R〉 = A) | |
2 | elreal 6727 | . . 3 ⊢ (B ∈ ℝ ↔ ∃y ∈ R 〈y, 0R〉 = B) | |
3 | breq1 3758 | . . . . . 6 ⊢ (〈x, 0R〉 = A → (〈x, 0R〉 <ℝ 〈y, 0R〉 ↔ A <ℝ 〈y, 0R〉)) | |
4 | breq2 3759 | . . . . . 6 ⊢ (〈x, 0R〉 = A → (〈y, 0R〉 <ℝ 〈x, 0R〉 ↔ 〈y, 0R〉 <ℝ A)) | |
5 | 3, 4 | orbi12d 706 | . . . . 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 591 | . . . 4 ⊢ (〈x, 0R〉 = A → (¬ (〈x, 0R〉 <ℝ 〈y, 0R〉 ∨ 〈y, 0R〉 <ℝ 〈x, 0R〉) ↔ ¬ (A <ℝ 〈y, 0R〉 ∨ 〈y, 0R〉 <ℝ A))) |
7 | eqeq1 2043 | . . . 4 ⊢ (〈x, 0R〉 = A → (〈x, 0R〉 = 〈y, 0R〉 ↔ A = 〈y, 0R〉)) | |
8 | 6, 7 | imbi12d 223 | . . 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 3759 | . . . . . 6 ⊢ (〈y, 0R〉 = B → (A <ℝ 〈y, 0R〉 ↔ A <ℝ B)) | |
10 | breq1 3758 | . . . . . 6 ⊢ (〈y, 0R〉 = B → (〈y, 0R〉 <ℝ A ↔ B <ℝ A)) | |
11 | 9, 10 | orbi12d 706 | . . . . 5 ⊢ (〈y, 0R〉 = B → ((A <ℝ 〈y, 0R〉 ∨ 〈y, 0R〉 <ℝ A) ↔ (A <ℝ B ∨ B <ℝ A))) |
12 | 11 | notbid 591 | . . . 4 ⊢ (〈y, 0R〉 = B → (¬ (A <ℝ 〈y, 0R〉 ∨ 〈y, 0R〉 <ℝ A) ↔ ¬ (A <ℝ B ∨ B <ℝ A))) |
13 | eqeq2 2046 | . . . 4 ⊢ (〈y, 0R〉 = B → (A = 〈y, 0R〉 ↔ A = B)) | |
14 | 12, 13 | imbi12d 223 | . . 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 6705 | . . . . 5 ⊢ ((x ∈ R ∧ y ∈ R ∧ ¬ (x <R y ∨ y <R x)) → x = y) | |
16 | 15 | 3expia 1105 | . . . 4 ⊢ ((x ∈ R ∧ y ∈ R) → (¬ (x <R y ∨ y <R x) → x = y)) |
17 | ltresr 6736 | . . . . . 6 ⊢ (〈x, 0R〉 <ℝ 〈y, 0R〉 ↔ x <R y) | |
18 | ltresr 6736 | . . . . . 6 ⊢ (〈y, 0R〉 <ℝ 〈x, 0R〉 ↔ y <R x) | |
19 | 17, 18 | orbi12i 680 | . . . . 5 ⊢ ((〈x, 0R〉 <ℝ 〈y, 0R〉 ∨ 〈y, 0R〉 <ℝ 〈x, 0R〉) ↔ (x <R y ∨ y <R x)) |
20 | 19 | notbii 593 | . . . 4 ⊢ (¬ (〈x, 0R〉 <ℝ 〈y, 0R〉 ∨ 〈y, 0R〉 <ℝ 〈x, 0R〉) ↔ ¬ (x <R y ∨ y <R x)) |
21 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
22 | 21 | eqresr 6733 | . . . 4 ⊢ (〈x, 0R〉 = 〈y, 0R〉 ↔ x = y) |
23 | 16, 20, 22 | 3imtr4g 194 | . . 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 2581 | . 2 ⊢ ((A ∈ ℝ ∧ B ∈ ℝ) → (¬ (A <ℝ B ∨ B <ℝ A) → A = B)) |
25 | 24 | 3impia 1100 | 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 97 ∨ wo 628 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 〈cop 3370 class class class wbr 3755 Rcnr 6281 0Rc0r 6282 <R cltr 6287 ℝcr 6710 <ℝ cltrr 6715 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-r 6721 df-lt 6724 |
This theorem is referenced by: (None) |
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