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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 15431 | . 2 ⊢ (ℵ‘1o) ≼ ℝ | |
2 | reex 10474 | . . . . 5 ⊢ ℝ ∈ V | |
3 | numth3 9738 | . . . . 5 ⊢ (ℝ ∈ V → ℝ ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ℝ ∈ dom card |
5 | nnenom 13198 | . . . . . . 7 ⊢ ℕ ≈ ω | |
6 | 5 | ensymi 8407 | . . . . . 6 ⊢ ω ≈ ℕ |
7 | ruc 15429 | . . . . . 6 ⊢ ℕ ≺ ℝ | |
8 | ensdomtr 8500 | . . . . . 6 ⊢ ((ω ≈ ℕ ∧ ℕ ≺ ℝ) → ω ≺ ℝ) | |
9 | 6, 7, 8 | mp2an 688 | . . . . 5 ⊢ ω ≺ ℝ |
10 | sdomdom 8385 | . . . . 5 ⊢ (ω ≺ ℝ → ω ≼ ℝ) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ ω ≼ ℝ |
12 | resdomq 15430 | . . . 4 ⊢ ℚ ≺ ℝ | |
13 | infdif 9477 | . . . 4 ⊢ ((ℝ ∈ dom card ∧ ω ≼ ℝ ∧ ℚ ≺ ℝ) → (ℝ ∖ ℚ) ≈ ℝ) | |
14 | 4, 11, 12, 13 | mp3an 1453 | . . 3 ⊢ (ℝ ∖ ℚ) ≈ ℝ |
15 | 14 | ensymi 8407 | . 2 ⊢ ℝ ≈ (ℝ ∖ ℚ) |
16 | domentr 8416 | . 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 2081 Vcvv 3437 ∖ cdif 3856 class class class wbr 4962 dom cdm 5443 ‘cfv 6225 ωcom 7436 1oc1o 7946 ≈ cen 8354 ≼ cdom 8355 ≺ csdm 8356 cardccrd 9210 ℵcale 9211 ℝcr 10382 ℕcn 11486 ℚcq 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-ac2 9731 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-omul 7958 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-oi 8820 df-har 8868 df-dju 9176 df-card 9214 df-aleph 9215 df-acn 9217 df-ac 9388 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-fz 12743 df-seq 13220 |
This theorem is referenced by: (None) |
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