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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 9486 | . . . . . 6 ⊢ (ℵ‘∅) = ω | |
2 | nnenom 13342 | . . . . . . 7 ⊢ ℕ ≈ ω | |
3 | 2 | ensymi 8553 | . . . . . 6 ⊢ ω ≈ ℕ |
4 | 1, 3 | eqbrtri 5080 | . . . . 5 ⊢ (ℵ‘∅) ≈ ℕ |
5 | ruc 15590 | . . . . 5 ⊢ ℕ ≺ ℝ | |
6 | ensdomtr 8647 | . . . . 5 ⊢ (((ℵ‘∅) ≈ ℕ ∧ ℕ ≺ ℝ) → (ℵ‘∅) ≺ ℝ) | |
7 | 4, 5, 6 | mp2an 690 | . . . 4 ⊢ (ℵ‘∅) ≺ ℝ |
8 | alephnbtwn2 9492 | . . . 4 ⊢ ¬ ((ℵ‘∅) ≺ ℝ ∧ ℝ ≺ (ℵ‘suc ∅)) | |
9 | 7, 8 | mptnan 1765 | . . 3 ⊢ ¬ ℝ ≺ (ℵ‘suc ∅) |
10 | df-1o 8096 | . . . . 5 ⊢ 1o = suc ∅ | |
11 | 10 | fveq2i 6668 | . . . 4 ⊢ (ℵ‘1o) = (ℵ‘suc ∅) |
12 | 11 | breq2i 5067 | . . 3 ⊢ (ℝ ≺ (ℵ‘1o) ↔ ℝ ≺ (ℵ‘suc ∅)) |
13 | 9, 12 | mtbir 325 | . 2 ⊢ ¬ ℝ ≺ (ℵ‘1o) |
14 | fvex 6678 | . . 3 ⊢ (ℵ‘1o) ∈ V | |
15 | reex 10622 | . . 3 ⊢ ℝ ∈ V | |
16 | domtri 9972 | . . 3 ⊢ (((ℵ‘1o) ∈ V ∧ ℝ ∈ V) → ((ℵ‘1o) ≼ ℝ ↔ ¬ ℝ ≺ (ℵ‘1o))) | |
17 | 14, 15, 16 | mp2an 690 | . 2 ⊢ ((ℵ‘1o) ≼ ℝ ↔ ¬ ℝ ≺ (ℵ‘1o)) |
18 | 13, 17 | mpbir 233 | 1 ⊢ (ℵ‘1o) ≼ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∈ wcel 2110 Vcvv 3495 ∅c0 4291 class class class wbr 5059 suc csuc 6188 ‘cfv 6350 ωcom 7574 1oc1o 8089 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 ℵcale 9359 ℝcr 10530 ℕcn 11632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-ac2 9879 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-har 9016 df-card 9362 df-aleph 9363 df-ac 9536 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 |
This theorem is referenced by: aleph1irr 15593 |
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