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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 9477 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
2 | nnenom 13343 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8542 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | 1, 3 | eqbrtri 5051 | . . . . 5 ⊢ (ℵ‘∅) ≈ ℕ |
5 | ruc 15588 | . . . . 5 ⊢ ℕ ≺ ℝ | |
6 | ensdomtr 8637 | . . . . 5 ⊢ (((ℵ‘∅) ≈ ℕ ∧ ℕ ≺ ℝ) → (ℵ‘∅) ≺ ℝ) | |
7 | 4, 5, 6 | mp2an 691 | . . . 4 ⊢ (ℵ‘∅) ≺ ℝ |
8 | alephnbtwn2 9483 | . . . 4 ⊢ ¬ ((ℵ‘∅) ≺ ℝ ∧ ℝ ≺ (ℵ‘suc ∅)) | |
9 | 7, 8 | mptnan 1770 | . . 3 ⊢ ¬ ℝ ≺ (ℵ‘suc ∅) |
10 | df-1o 8085 | . . . . 5 ⊢ 1o = suc ∅ | |
11 | 10 | fveq2i 6648 | . . . 4 ⊢ (ℵ‘1o) = (ℵ‘suc ∅) |
12 | 11 | breq2i 5038 | . . 3 ⊢ (ℝ ≺ (ℵ‘1o) ↔ ℝ ≺ (ℵ‘suc ∅)) |
13 | 9, 12 | mtbir 326 | . 2 ⊢ ¬ ℝ ≺ (ℵ‘1o) |
14 | fvex 6658 | . . 3 ⊢ (ℵ‘1o) ∈ V | |
15 | reex 10617 | . . 3 ⊢ ℝ ∈ V | |
16 | domtri 9967 | . . 3 ⊢ (((ℵ‘1o) ∈ V ∧ ℝ ∈ V) → ((ℵ‘1o) ≼ ℝ ↔ ¬ ℝ ≺ (ℵ‘1o))) | |
17 | 14, 15, 16 | mp2an 691 | . 2 ⊢ ((ℵ‘1o) ≼ ℝ ↔ ¬ ℝ ≺ (ℵ‘1o)) |
18 | 13, 17 | mpbir 234 | 1 ⊢ (ℵ‘1o) ≼ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 class class class wbr 5030 suc csuc 6161 ‘cfv 6324 ωcom 7560 1oc1o 8078 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 ℵcale 9349 ℝcr 10525 ℕcn 11625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-ac2 9874 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-oi 8958 df-har 9005 df-card 9352 df-aleph 9353 df-ac 9527 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 |
This theorem is referenced by: aleph1irr 15591 |
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