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Mirrors > Home > MPE Home > Th. List > gch3 | Structured version Visualization version GIF version |
Description: An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
gch3 | ⊢ (GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → 𝑥 ∈ On) | |
2 | fvex 6682 | . . . . 5 ⊢ (ℵ‘𝑥) ∈ V | |
3 | simpl 485 | . . . . 5 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → GCH = V) | |
4 | 2, 3 | eleqtrrid 2920 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH) |
5 | fvex 6682 | . . . . 5 ⊢ (ℵ‘suc 𝑥) ∈ V | |
6 | 5, 3 | eleqtrrid 2920 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH) |
7 | gchaleph2 10093 | . . . 4 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) | |
8 | 1, 4, 6, 7 | syl3anc 1367 | . . 3 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
9 | 8 | ralrimiva 3182 | . 2 ⊢ (GCH = V → ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
10 | alephgch 10095 | . . . . . 6 ⊢ ((ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → (ℵ‘𝑥) ∈ GCH) | |
11 | 10 | ralimi 3160 | . . . . 5 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
12 | alephfnon 9490 | . . . . . 6 ⊢ ℵ Fn On | |
13 | ffnfv 6881 | . . . . . 6 ⊢ (ℵ:On⟶GCH ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH)) | |
14 | 12, 13 | mpbiran 707 | . . . . 5 ⊢ (ℵ:On⟶GCH ↔ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
15 | 11, 14 | sylibr 236 | . . . 4 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ℵ:On⟶GCH) |
16 | 15 | frnd 6520 | . . 3 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ran ℵ ⊆ GCH) |
17 | gch2 10096 | . . 3 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH) | |
18 | 16, 17 | sylibr 236 | . 2 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → GCH = V) |
19 | 9, 18 | impbii 211 | 1 ⊢ (GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5065 ran crn 5555 Oncon0 6190 suc csuc 6192 Fn wfn 6349 ⟶wf 6350 ‘cfv 6354 ≈ cen 8505 ℵcale 9364 GCHcgch 10041 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-reg 9055 ax-inf2 9103 |
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-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-seqom 8083 df-1o 8101 df-2o 8102 df-oadd 8105 df-omul 8106 df-oexp 8107 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-oi 8973 df-har 9021 df-wdom 9022 df-cnf 9124 df-r1 9192 df-rank 9193 df-dju 9329 df-card 9367 df-aleph 9368 df-ac 9541 df-fin4 9708 df-gch 10042 |
This theorem is referenced by: (None) |
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