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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 15586 | . 2 ⊢ (ℵ‘1o) ≼ ℝ | |
2 | reex 10616 | . . . . 5 ⊢ ℝ ∈ V | |
3 | numth3 9880 | . . . . 5 ⊢ (ℝ ∈ V → ℝ ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℝ ∈ dom card |
5 | nnenom 13336 | . . . . . . 7 ⊢ ℕ ≈ ω | |
6 | 5 | ensymi 8547 | . . . . . 6 ⊢ ω ≈ ℕ |
7 | ruc 15584 | . . . . . 6 ⊢ ℕ ≺ ℝ | |
8 | ensdomtr 8641 | . . . . . 6 ⊢ ((ω ≈ ℕ ∧ ℕ ≺ ℝ) → ω ≺ ℝ) | |
9 | 6, 7, 8 | mp2an 688 | . . . . 5 ⊢ ω ≺ ℝ |
10 | sdomdom 8525 | . . . . 5 ⊢ (ω ≺ ℝ → ω ≼ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ω ≼ ℝ |
12 | resdomq 15585 | . . . 4 ⊢ ℚ ≺ ℝ | |
13 | infdif 9619 | . . . 4 ⊢ ((ℝ ∈ dom card ∧ ω ≼ ℝ ∧ ℚ ≺ ℝ) → (ℝ ∖ ℚ) ≈ ℝ) | |
14 | 4, 11, 12, 13 | mp3an 1452 | . . 3 ⊢ (ℝ ∖ ℚ) ≈ ℝ |
15 | 14 | ensymi 8547 | . 2 ⊢ ℝ ≈ (ℝ ∖ ℚ) |
16 | domentr 8556 | . 2 ⊢ (((ℵ‘1o) ≼ ℝ ∧ ℝ ≈ (ℝ ∖ ℚ)) → (ℵ‘1o) ≼ (ℝ ∖ ℚ)) | |
17 | 1, 15, 16 | mp2an 688 | 1 ⊢ (ℵ‘1o) ≼ (ℝ ∖ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 ωcom 7569 1oc1o 8084 ≈ cen 8494 ≼ cdom 8495 ≺ csdm 8496 cardccrd 9352 ℵcale 9353 ℝcr 10524 ℕcn 11626 ℚcq 12336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-ac2 9873 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-har 9010 df-dju 9318 df-card 9356 df-aleph 9357 df-acn 9359 df-ac 9530 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-fz 12881 df-seq 13358 |
This theorem is referenced by: (None) |
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