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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 25640. (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 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
2 | 1 | fconst6 6008 | . . 3 ⊢ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ |
3 | elee 25574 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {1}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {1}):(1...𝑁)⟶ℝ)) | |
4 | 2, 3 | mpbiri 247 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ∈ (𝔼‘𝑁)) |
5 | 0re 9919 | . . . 4 ⊢ 0 ∈ ℝ | |
6 | 5 | fconst6 6008 | . . 3 ⊢ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ |
7 | elee 25574 | . . 3 ⊢ (𝑁 ∈ ℕ → (((1...𝑁) × {0}) ∈ (𝔼‘𝑁) ↔ ((1...𝑁) × {0}):(1...𝑁)⟶ℝ)) | |
8 | 6, 7 | mpbiri 247 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {0}) ∈ (𝔼‘𝑁)) |
9 | ax-1ne0 9884 | . . . . . . 7 ⊢ 1 ≠ 0 | |
10 | 9 | neii 2784 | . . . . . 6 ⊢ ¬ 1 = 0 |
11 | 1ex 9914 | . . . . . . 7 ⊢ 1 ∈ V | |
12 | 11 | sneqr 4311 | . . . . . 6 ⊢ ({1} = {0} → 1 = 0) |
13 | 10, 12 | mto 187 | . . . . 5 ⊢ ¬ {1} = {0} |
14 | elnnuz 11600 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
15 | eluzfz1 12219 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘1) → 1 ∈ (1...𝑁)) | |
16 | 14, 15 | sylbi 206 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → 1 ∈ (1...𝑁)) |
17 | ne0i 3880 | . . . . . . . 8 ⊢ (1 ∈ (1...𝑁) → (1...𝑁) ≠ ∅) | |
18 | 16, 17 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → (1...𝑁) ≠ ∅) |
19 | rnxp 5483 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {1}) = {1}) | |
20 | 18, 19 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {1}) = {1}) |
21 | rnxp 5483 | . . . . . . 7 ⊢ ((1...𝑁) ≠ ∅ → ran ((1...𝑁) × {0}) = {0}) | |
22 | 18, 21 | syl 17 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → ran ((1...𝑁) × {0}) = {0}) |
23 | 20, 22 | eqeq12d 2625 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0}) ↔ {1} = {0})) |
24 | 13, 23 | mtbiri 316 | . . . 4 ⊢ (𝑁 ∈ ℕ → ¬ ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) |
25 | rneq 5272 | . . . 4 ⊢ (((1...𝑁) × {1}) = ((1...𝑁) × {0}) → ran ((1...𝑁) × {1}) = ran ((1...𝑁) × {0})) | |
26 | 24, 25 | nsyl 134 | . . 3 ⊢ (𝑁 ∈ ℕ → ¬ ((1...𝑁) × {1}) = ((1...𝑁) × {0})) |
27 | 26 | neqned 2789 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0})) |
28 | neeq1 2844 | . . 3 ⊢ (𝑥 = ((1...𝑁) × {1}) → (𝑥 ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ 𝑦)) | |
29 | neeq2 2845 | . . 3 ⊢ (𝑦 = ((1...𝑁) × {0}) → (((1...𝑁) × {1}) ≠ 𝑦 ↔ ((1...𝑁) × {1}) ≠ ((1...𝑁) × {0}))) | |
30 | 28, 29 | rspc2ev 3295 | . 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) |
31 | 4, 8, 27, 30 | syl3anc 1318 | 1 ⊢ (𝑁 ∈ ℕ → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑦 ∈ (𝔼‘𝑁)𝑥 ≠ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∅c0 3874 {csn 4125 × cxp 5036 ran crn 5039 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 ℕcn 10897 ℤ≥cuz 11563 ...cfz 12197 𝔼cee 25568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-z 11255 df-uz 11564 df-fz 12198 df-ee 25571 |
This theorem is referenced by: btwndiff 31304 |
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