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Mirrors > Home > MPE Home > Th. List > alephsucpw2 | Structured version Visualization version GIF version |
Description: The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 10416 or gchaleph2 10412.) The transposed form alephsucpw 10310 cannot be proven without the AC, and is in fact equivalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.) |
Ref | Expression |
---|---|
alephsucpw2 | ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6781 | . . 3 ⊢ (ℵ‘𝐴) ∈ V | |
2 | 1 | canth2 8882 | . 2 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) |
3 | alephnbtwn2 9812 | . 2 ⊢ ¬ ((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)) | |
4 | 2, 3 | mptnan 1774 | 1 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 𝒫 cpw 4538 class class class wbr 5078 suc csuc 6265 ‘cfv 6430 ≺ csdm 8706 ℵcale 9678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-oi 9230 df-har 9277 df-card 9681 df-aleph 9682 |
This theorem is referenced by: alephsucpw 10310 gchaleph 10411 |
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