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Mirrors > Home > MPE Home > Th. List > aleph1re | Structured version Visualization version GIF version |
Description: There are at least aleph-one real numbers. (Contributed by NM, 2-Feb-2005.) |
Ref | Expression |
---|---|
aleph1re | ⊢ (ℵ‘1𝑜) ≼ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aleph0 9089 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
2 | nnenom 12987 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8159 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | 1, 3 | eqbrtri 4807 | . . . . 5 ⊢ (ℵ‘∅) ≈ ℕ |
5 | ruc 15178 | . . . . 5 ⊢ ℕ ≺ ℝ | |
6 | ensdomtr 8252 | . . . . 5 ⊢ (((ℵ‘∅) ≈ ℕ ∧ ℕ ≺ ℝ) → (ℵ‘∅) ≺ ℝ) | |
7 | 4, 5, 6 | mp2an 672 | . . . 4 ⊢ (ℵ‘∅) ≺ ℝ |
8 | alephnbtwn2 9095 | . . . 4 ⊢ ¬ ((ℵ‘∅) ≺ ℝ ∧ ℝ ≺ (ℵ‘suc ∅)) | |
9 | 7, 8 | mptnan 1841 | . . 3 ⊢ ¬ ℝ ≺ (ℵ‘suc ∅) |
10 | df-1o 7713 | . . . . 5 ⊢ 1𝑜 = suc ∅ | |
11 | 10 | fveq2i 6335 | . . . 4 ⊢ (ℵ‘1𝑜) = (ℵ‘suc ∅) |
12 | 11 | breq2i 4794 | . . 3 ⊢ (ℝ ≺ (ℵ‘1𝑜) ↔ ℝ ≺ (ℵ‘suc ∅)) |
13 | 9, 12 | mtbir 312 | . 2 ⊢ ¬ ℝ ≺ (ℵ‘1𝑜) |
14 | fvex 6342 | . . 3 ⊢ (ℵ‘1𝑜) ∈ V | |
15 | reex 10229 | . . 3 ⊢ ℝ ∈ V | |
16 | domtri 9580 | . . 3 ⊢ (((ℵ‘1𝑜) ∈ V ∧ ℝ ∈ V) → ((ℵ‘1𝑜) ≼ ℝ ↔ ¬ ℝ ≺ (ℵ‘1𝑜))) | |
17 | 14, 15, 16 | mp2an 672 | . 2 ⊢ ((ℵ‘1𝑜) ≼ ℝ ↔ ¬ ℝ ≺ (ℵ‘1𝑜)) |
18 | 13, 17 | mpbir 221 | 1 ⊢ (ℵ‘1𝑜) ≼ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∈ wcel 2145 Vcvv 3351 ∅c0 4063 class class class wbr 4786 suc csuc 5868 ‘cfv 6031 ωcom 7212 1𝑜c1o 7706 ≈ cen 8106 ≼ cdom 8107 ≺ csdm 8108 ℵcale 8962 ℝcr 10137 ℕcn 11222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-ac2 9487 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-oi 8571 df-har 8619 df-card 8965 df-aleph 8966 df-ac 9139 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-seq 13009 |
This theorem is referenced by: aleph1irr 15181 |
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