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Mirrors > Home > MPE Home > Th. List > alephsucpw | Structured version Visualization version GIF version |
Description: The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10188 or gchaleph2 10184.) (Contributed by NM, 27-Aug-2005.) |
Ref | Expression |
---|---|
alephsucpw | ⊢ (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alephsucpw2 9623 | . 2 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) | |
2 | fvex 6699 | . . 3 ⊢ (ℵ‘suc 𝐴) ∈ V | |
3 | fvex 6699 | . . . 4 ⊢ (ℵ‘𝐴) ∈ V | |
4 | 3 | pwex 5257 | . . 3 ⊢ 𝒫 (ℵ‘𝐴) ∈ V |
5 | domtri 10068 | . . 3 ⊢ (((ℵ‘suc 𝐴) ∈ V ∧ 𝒫 (ℵ‘𝐴) ∈ V) → ((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ↔ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))) | |
6 | 2, 4, 5 | mp2an 692 | . 2 ⊢ ((ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) ↔ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) |
7 | 1, 6 | mpbir 234 | 1 ⊢ (ℵ‘suc 𝐴) ≼ 𝒫 (ℵ‘𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∈ wcel 2114 Vcvv 3400 𝒫 cpw 4498 class class class wbr 5040 suc csuc 6184 ‘cfv 6349 ≼ cdom 8565 ≺ csdm 8566 ℵcale 9450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-ac2 9975 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-om 7612 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-oi 9059 df-har 9106 df-card 9453 df-aleph 9454 df-ac 9628 |
This theorem is referenced by: aleph1 10083 |
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