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| Mirrors > Home > MPE Home > Th. List > axlowdim1 | Structured version Visualization version GIF version | ||
| Description: The lower dimension axiom for one dimension. In any dimension, there are at least two distinct points. Theorem 3.13 of [Schwabhauser] p. 32, where it is derived from axlowdim2 28975. (Contributed by Scott Fenton, 22-Apr-2013.) |
| Ref | Expression |
|---|---|
| axlowdim1 | ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 2 | 1 | fconst6 6798 | . . 3 ⊢ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ |
| 3 | elee 28909 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {1}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ)) | |
| 4 | 2, 3 | mpbiri 258 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ∈ (𝔼‘𝑁)) |
| 5 | 0re 11263 | . . . 4 ⊢ 0 ∈ ℝ | |
| 6 | 5 | fconst6 6798 | . . 3 ⊢ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ |
| 7 | elee 28909 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {0}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ)) | |
| 8 | 6, 7 | mpbiri 258 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {0}) ∈ (𝔼‘𝑁)) |
| 9 | ax-1ne0 11224 | . . . . . . 7 ⊢ 1 ≠ 0 | |
| 10 | 9 | neii 2942 | . . . . . 6 ⊢ ¬ 1 = 0 |
| 11 | 1ex 11257 | . . . . . . 7 ⊢ 1 ∈ V | |
| 12 | 11 | sneqr 4840 | . . . . . 6 ⊢ ({1} = {0} → 1 = 0) |
| 13 | 10, 12 | mto 197 | . . . . 5 ⊢ ¬ {1} = {0} |
| 14 | elnnuz 12922 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
| 15 | eluzfz1 13571 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
| 16 | 14, 15 | sylbi 217 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁)) |
| 17 | 16 | ne0d 4342 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
| 18 | rnxp 6190 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {1}) = {1}) | |
| 19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {1}) = {1}) |
| 20 | rnxp 6190 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {0}) = {0}) | |
| 21 | 17, 20 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {0}) = {0}) |
| 22 | 19, 21 | eqeq12d 2753 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0}) ↔ {1} = {0})) |
| 23 | 13, 22 | mtbiri 327 | . . . 4 ⊢ (𝑁 ∈ ℕ → ¬ ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) |
| 24 | rneq 5947 | . . . 4 ⊢ (((1...𝑁) × {1}) = ((1...𝑁) × {0}) → ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) | |
| 25 | 23, 24 | nsyl 140 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ ((1...𝑁) × {1}) = ((1...𝑁) × {0})) |
| 26 | 25 | neqned 2947 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) |
| 27 | neeq1 3003 | . . 3 ⊢ (𝑥 = ((1...𝑁) × {1}) → (𝑥 ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ 𝑦)) | |
| 28 | neeq2 3004 | . . 3 ⊢ (𝑦 = ((1...𝑁) × {0}) → (((1...𝑁) × {1}) ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0}))) | |
| 29 | 27, 28 | rspc2ev 3635 | . 2 ⊢ ((((1...𝑁) × {1}) ∈ (𝔼‘𝑁) ∧ ((1...𝑁) × {0}) ∈ (𝔼‘𝑁) ∧ ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
| 30 | 4, 8, 26, 29 | syl3anc 1373 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∅c0 4333 {csn 4626 × cxp 5683 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 ℕcn 12266 ℤ≥cuz 12878 ...cfz 13547 𝔼cee 28903 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-z 12614 df-uz 12879 df-fz 13548 df-ee 28906 |
| This theorem is referenced by: btwndiff 36028 |
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