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| Mirrors > Home > MPE Home > Th. List > ruc | Structured version Visualization version GIF version | ||
| Description: The set of positive integers is strictly dominated by the set of real numbers, i.e. the real numbers are uncountable. The proof consists of lemmas ruclem1 16263 through ruclem13 16274 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 16274 for the function existence version of this theorem. For an informal discussion of this proof, see mmcomplex.html#uncountable 16274. For an alternate proof see rucALT 16262. This is Metamath 100 proof #22. (Contributed by NM, 13-Oct-2004.) |
| Ref | Expression |
|---|---|
| ruc | ⊢ ℕ ≺ ℝ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reex 11242 | . . 3 ⊢ ℝ ∈ V | |
| 2 | nnssre 12266 | . . 3 ⊢ ℕ ⊆ ℝ | |
| 3 | ssdomg 9036 | . . 3 ⊢ (ℝ ∈ V → (ℕ ⊆ ℝ → ℕ ≼ ℝ)) | |
| 4 | 1, 2, 3 | mp2 9 | . 2 ⊢ ℕ ≼ ℝ |
| 5 | ruclem13 16274 | . . . . 5 ⊢ ¬ 𝑓:ℕ–onto→ℝ | |
| 6 | f1ofo 6853 | . . . . 5 ⊢ (𝑓:ℕ–1-1-onto→ℝ → 𝑓:ℕ–onto→ℝ) | |
| 7 | 5, 6 | mto 197 | . . . 4 ⊢ ¬ 𝑓:ℕ–1-1-onto→ℝ |
| 8 | 7 | nex 1800 | . . 3 ⊢ ¬ ∃𝑓 𝑓:ℕ–1-1-onto→ℝ |
| 9 | bren 8991 | . . 3 ⊢ (ℕ ≈ ℝ ↔ ∃𝑓 𝑓:ℕ–1-1-onto→ℝ) | |
| 10 | 8, 9 | mtbir 323 | . 2 ⊢ ¬ ℕ ≈ ℝ |
| 11 | brsdom 9011 | . 2 ⊢ (ℕ ≺ ℝ ↔ (ℕ ≼ ℝ ∧ ¬ ℕ ≈ ℝ)) | |
| 12 | 4, 10, 11 | mpbir2an 711 | 1 ⊢ ℕ ≺ ℝ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∃wex 1779 ∈ wcel 2108 Vcvv 3479 ⊆ wss 3950 class class class wbr 5141 –onto→wfo 6557 –1-1-onto→wf1o 6558 ≈ cen 8978 ≼ cdom 8979 ≺ csdm 8980 ℝcr 11150 ℕcn 12262 |
| 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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 |
| 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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-sup 9478 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-n0 12523 df-z 12610 df-uz 12875 df-fz 13544 df-seq 14039 |
| This theorem is referenced by: resdomq 16276 aleph1re 16277 aleph1irr 16278 |
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