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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 15578 through ruclem13 15589 and this final piece. Our proof is based on the proof of Theorem 5.18 of [Truss] p. 114. See ruclem13 15589 for the function existence version of this theorem. For an informal discussion of this proof, see mmcomplex.html#uncountable 15589. For an alternate proof see rucALT 15577. This is Metamath 100 proof #22. (Contributed by NM, 13-Oct-2004.) |
Ref | Expression |
---|---|
ruc | ⊢ ℕ ≺ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 10622 | . . 3 ⊢ ℝ ∈ V | |
2 | nnssre 11636 | . . 3 ⊢ ℕ ⊆ ℝ | |
3 | ssdomg 8549 | . . 3 ⊢ (ℝ ∈ V → (ℕ ⊆ ℝ → ℕ ≼ ℝ)) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ ℕ ≼ ℝ |
5 | ruclem13 15589 | . . . . 5 ⊢ ¬ 𝑓:ℕ–onto→ℝ | |
6 | f1ofo 6616 | . . . . 5 ⊢ (𝑓:ℕ–1-1-onto→ℝ → 𝑓:ℕ–onto→ℝ) | |
7 | 5, 6 | mto 199 | . . . 4 ⊢ ¬ 𝑓:ℕ–1-1-onto→ℝ |
8 | 7 | nex 1797 | . . 3 ⊢ ¬ ∃𝑓 𝑓:ℕ–1-1-onto→ℝ |
9 | bren 8512 | . . 3 ⊢ (ℕ ≈ ℝ ↔ ∃𝑓 𝑓:ℕ–1-1-onto→ℝ) | |
10 | 8, 9 | mtbir 325 | . 2 ⊢ ¬ ℕ ≈ ℝ |
11 | brsdom 8526 | . 2 ⊢ (ℕ ≺ ℝ ↔ (ℕ ≼ ℝ ∧ ¬ ℕ ≈ ℝ)) | |
12 | 4, 10, 11 | mpbir2an 709 | 1 ⊢ ℕ ≺ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 ⊆ wss 3935 class class class wbr 5058 –onto→wfo 6347 –1-1-onto→wf1o 6348 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 ℝcr 10530 ℕcn 11632 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 |
This theorem is referenced by: resdomq 15591 aleph1re 15592 aleph1irr 15593 |
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