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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 16287 | . 2 ⊢ (ℵ‘1o) ≼ ℝ | |
2 | reex 11271 | . . . . 5 ⊢ ℝ ∈ V | |
3 | numth3 10535 | . . . . 5 ⊢ (ℝ ∈ V → ℝ ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℝ ∈ dom card |
5 | nnenom 14027 | . . . . . . 7 ⊢ ℕ ≈ ω | |
6 | 5 | ensymi 9060 | . . . . . 6 ⊢ ω ≈ ℕ |
7 | ruc 16285 | . . . . . 6 ⊢ ℕ ≺ ℝ | |
8 | ensdomtr 9175 | . . . . . 6 ⊢ ((ω ≈ ℕ ∧ ℕ ≺ ℝ) → ω ≺ ℝ) | |
9 | 6, 7, 8 | mp2an 691 | . . . . 5 ⊢ ω ≺ ℝ |
10 | sdomdom 9036 | . . . . 5 ⊢ (ω ≺ ℝ → ω ≼ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ω ≼ ℝ |
12 | resdomq 16286 | . . . 4 ⊢ ℚ ≺ ℝ | |
13 | infdif 10273 | . . . 4 ⊢ ((ℝ ∈ dom card ∧ ω ≼ ℝ ∧ ℚ ≺ ℝ) → (ℝ ∖ ℚ) ≈ ℝ) | |
14 | 4, 11, 12, 13 | mp3an 1461 | . . 3 ⊢ (ℝ ∖ ℚ) ≈ ℝ |
15 | 14 | ensymi 9060 | . 2 ⊢ ℝ ≈ (ℝ ∖ ℚ) |
16 | domentr 9069 | . 2 ⊢ (((ℵ‘1o) ≼ ℝ ∧ ℝ ≈ (ℝ ∖ ℚ)) → (ℵ‘1o) ≼ (ℝ ∖ ℚ)) | |
17 | 1, 15, 16 | mp2an 691 | 1 ⊢ (ℵ‘1o) ≼ (ℝ ∖ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2103 Vcvv 3482 ∖ cdif 3967 class class class wbr 5169 dom cdm 5699 ‘cfv 6572 ωcom 7899 1oc1o 8511 ≈ cen 8996 ≼ cdom 8997 ≺ csdm 8998 cardccrd 10000 ℵcale 10001 ℝcr 11179 ℕcn 12289 ℚcq 13009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 ax-ac2 10528 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-oadd 8522 df-omul 8523 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-oi 9575 df-har 9622 df-dju 9966 df-card 10004 df-aleph 10005 df-acn 10007 df-ac 10181 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-n0 12550 df-z 12636 df-uz 12900 df-q 13010 df-fz 13564 df-seq 14049 |
This theorem is referenced by: (None) |
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