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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 14974 | . 2 ⊢ (ℵ‘1𝑜) ≼ ℝ | |
2 | reex 10027 | . . . . 5 ⊢ ℝ ∈ V | |
3 | numth3 9292 | . . . . 5 ⊢ (ℝ ∈ V → ℝ ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℝ ∈ dom card |
5 | nnenom 12779 | . . . . . . 7 ⊢ ℕ ≈ ω | |
6 | 5 | ensymi 8006 | . . . . . 6 ⊢ ω ≈ ℕ |
7 | ruc 14972 | . . . . . 6 ⊢ ℕ ≺ ℝ | |
8 | ensdomtr 8096 | . . . . . 6 ⊢ ((ω ≈ ℕ ∧ ℕ ≺ ℝ) → ω ≺ ℝ) | |
9 | 6, 7, 8 | mp2an 708 | . . . . 5 ⊢ ω ≺ ℝ |
10 | sdomdom 7983 | . . . . 5 ⊢ (ω ≺ ℝ → ω ≼ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ω ≼ ℝ |
12 | resdomq 14973 | . . . 4 ⊢ ℚ ≺ ℝ | |
13 | infdif 9031 | . . . 4 ⊢ ((ℝ ∈ dom card ∧ ω ≼ ℝ ∧ ℚ ≺ ℝ) → (ℝ ∖ ℚ) ≈ ℝ) | |
14 | 4, 11, 12, 13 | mp3an 1424 | . . 3 ⊢ (ℝ ∖ ℚ) ≈ ℝ |
15 | 14 | ensymi 8006 | . 2 ⊢ ℝ ≈ (ℝ ∖ ℚ) |
16 | domentr 8015 | . 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 1990 Vcvv 3200 ∖ cdif 3571 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 ωcom 7065 1𝑜c1o 7553 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 cardccrd 8761 ℵcale 8762 ℝcr 9935 ℕcn 11020 ℚcq 11788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-har 8463 df-card 8765 df-aleph 8766 df-acn 8768 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-fz 12327 df-seq 12802 |
This theorem is referenced by: (None) |
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