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Mirrors > Home > MPE Home > Th. List > aleph1irr | Structured version Visualization version GIF version |
Description: There are at least aleph-one irrationals. (Contributed by NM, 2-Feb-2005.) |
Ref | Expression |
---|---|
aleph1irr | ⊢ (ℵ‘1𝑜) ≼ (ℝ ∖ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aleph1re 15018 | . 2 ⊢ (ℵ‘1𝑜) ≼ ℝ | |
2 | reex 10065 | . . . . 5 ⊢ ℝ ∈ V | |
3 | numth3 9330 | . . . . 5 ⊢ (ℝ ∈ V → ℝ ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℝ ∈ dom card |
5 | nnenom 12819 | . . . . . . 7 ⊢ ℕ ≈ ω | |
6 | 5 | ensymi 8047 | . . . . . 6 ⊢ ω ≈ ℕ |
7 | ruc 15016 | . . . . . 6 ⊢ ℕ ≺ ℝ | |
8 | ensdomtr 8137 | . . . . . 6 ⊢ ((ω ≈ ℕ ∧ ℕ ≺ ℝ) → ω ≺ ℝ) | |
9 | 6, 7, 8 | mp2an 708 | . . . . 5 ⊢ ω ≺ ℝ |
10 | sdomdom 8025 | . . . . 5 ⊢ (ω ≺ ℝ → ω ≼ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ω ≼ ℝ |
12 | resdomq 15017 | . . . 4 ⊢ ℚ ≺ ℝ | |
13 | infdif 9069 | . . . 4 ⊢ ((ℝ ∈ dom card ∧ ω ≼ ℝ ∧ ℚ ≺ ℝ) → (ℝ ∖ ℚ) ≈ ℝ) | |
14 | 4, 11, 12, 13 | mp3an 1464 | . . 3 ⊢ (ℝ ∖ ℚ) ≈ ℝ |
15 | 14 | ensymi 8047 | . 2 ⊢ ℝ ≈ (ℝ ∖ ℚ) |
16 | domentr 8056 | . 2 ⊢ (((ℵ‘1𝑜) ≼ ℝ ∧ ℝ ≈ (ℝ ∖ ℚ)) → (ℵ‘1𝑜) ≼ (ℝ ∖ ℚ)) | |
17 | 1, 15, 16 | mp2an 708 | 1 ⊢ (ℵ‘1𝑜) ≼ (ℝ ∖ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 Vcvv 3231 ∖ cdif 3604 class class class wbr 4685 dom cdm 5143 ‘cfv 5926 ωcom 7107 1𝑜c1o 7598 ≈ cen 7994 ≼ cdom 7995 ≺ csdm 7996 cardccrd 8799 ℵcale 8800 ℝcr 9973 ℕcn 11058 ℚcq 11826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-ac2 9323 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-omul 7610 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-oi 8456 df-har 8504 df-card 8803 df-aleph 8804 df-acn 8806 df-ac 8977 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-uz 11726 df-q 11827 df-fz 12365 df-seq 12842 |
This theorem is referenced by: (None) |
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