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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 28990. (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 11259 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 1 | fconst6 6799 | . . 3 ⊢ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ |
3 | elee 28924 | . . 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 11261 | . . . 4 ⊢ 0 ∈ ℝ | |
6 | 5 | fconst6 6799 | . . 3 ⊢ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ |
7 | elee 28924 | . . 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 11222 | . . . . . . 7 ⊢ 1 ≠ 0 | |
10 | 9 | neii 2940 | . . . . . 6 ⊢ ¬ 1 = 0 |
11 | 1ex 11255 | . . . . . . 7 ⊢ 1 ∈ V | |
12 | 11 | sneqr 4845 | . . . . . 6 ⊢ ({1} = {0} → 1 = 0) |
13 | 10, 12 | mto 197 | . . . . 5 ⊢ ¬ {1} = {0} |
14 | elnnuz 12920 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
15 | eluzfz1 13568 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
16 | 14, 15 | sylbi 217 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁)) |
17 | 16 | ne0d 4348 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
18 | rnxp 6192 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {1}) = {1}) | |
19 | 17, 18 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {1}) = {1}) |
20 | rnxp 6192 | . . . . . . 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 2751 | . . . . 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 5950 | . . . 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 2945 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) |
27 | neeq1 3001 | . . 3 ⊢ (𝑥 = ((1...𝑁) × {1}) → (𝑥 ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ 𝑦)) | |
28 | neeq2 3002 | . . 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 1370 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 ∅c0 4339 {csn 4631 × cxp 5687 ran crn 5690 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 ℕcn 12264 ℤ≥cuz 12876 ...cfz 13544 𝔼cee 28918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-z 12612 df-uz 12877 df-fz 13545 df-ee 28921 |
This theorem is referenced by: btwndiff 36009 |
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