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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 26754. (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 10630 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 1 | fconst6 6543 | . . 3 ⊢ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ |
3 | elee 26688 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {1}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ)) | |
4 | 2, 3 | mpbiri 261 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ∈ (𝔼‘𝑁)) |
5 | 0re 10632 | . . . 4 ⊢ 0 ∈ ℝ | |
6 | 5 | fconst6 6543 | . . 3 ⊢ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ |
7 | elee 26688 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {0}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ)) | |
8 | 6, 7 | mpbiri 261 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {0}) ∈ (𝔼‘𝑁)) |
9 | ax-1ne0 10595 | . . . . . . 7 ⊢ 1 ≠ 0 | |
10 | 9 | neii 2989 | . . . . . 6 ⊢ ¬ 1 = 0 |
11 | 1ex 10626 | . . . . . . 7 ⊢ 1 ∈ V | |
12 | 11 | sneqr 4731 | . . . . . 6 ⊢ ({1} = {0} → 1 = 0) |
13 | 10, 12 | mto 200 | . . . . 5 ⊢ ¬ {1} = {0} |
14 | elnnuz 12270 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
15 | eluzfz1 12909 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
16 | 14, 15 | sylbi 220 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁)) |
17 | 16 | ne0d 4251 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
18 | rnxp 5994 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {1}) = {1}) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {1}) = {1}) |
20 | rnxp 5994 | . . . . . . 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 2814 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0}) ↔ {1} = {0})) |
23 | 13, 22 | mtbiri 330 | . . . 4 ⊢ (𝑁 ∈ ℕ → ¬ ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) |
24 | rneq 5770 | . . . 4 ⊢ (((1...𝑁) × {1}) = ((1...𝑁) × {0}) → ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) | |
25 | 23, 24 | nsyl 142 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ ((1...𝑁) × {1}) = ((1...𝑁) × {0})) |
26 | 25 | neqned 2994 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) |
27 | neeq1 3049 | . . 3 ⊢ (𝑥 = ((1...𝑁) × {1}) → (𝑥 ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ 𝑦)) | |
28 | neeq2 3050 | . . 3 ⊢ (𝑦 = ((1...𝑁) × {0}) → (((1...𝑁) × {1}) ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0}))) | |
29 | 27, 28 | rspc2ev 3583 | . 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 1368 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ∅c0 4243 {csn 4525 × cxp 5517 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 ℕcn 11625 ℤ≥cuz 12231 ...cfz 12885 𝔼cee 26682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-z 11970 df-uz 12232 df-fz 12886 df-ee 26685 |
This theorem is referenced by: btwndiff 33601 |
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