![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 479 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → 𝑥 ∈ On) | |
2 | fvex 6459 | . . . . 5 ⊢ (ℵ‘𝑥) ∈ V | |
3 | simpl 476 | . . . . 5 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → GCH = V) | |
4 | 2, 3 | syl5eleqr 2865 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH) |
5 | fvex 6459 | . . . . 5 ⊢ (ℵ‘suc 𝑥) ∈ V | |
6 | 5, 3 | syl5eleqr 2865 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH) |
7 | gchaleph2 9829 | . . . 4 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) | |
8 | 1, 4, 6, 7 | syl3anc 1439 | . . 3 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
9 | 8 | ralrimiva 3147 | . 2 ⊢ (GCH = V → ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
10 | alephgch 9831 | . . . . . 6 ⊢ ((ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → (ℵ‘𝑥) ∈ GCH) | |
11 | 10 | ralimi 3133 | . . . . 5 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
12 | alephfnon 9221 | . . . . . 6 ⊢ ℵ Fn On | |
13 | ffnfv 6652 | . . . . . 6 ⊢ (ℵ:On⟶GCH ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH)) | |
14 | 12, 13 | mpbiran 699 | . . . . 5 ⊢ (ℵ:On⟶GCH ↔ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
15 | 11, 14 | sylibr 226 | . . . 4 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ℵ:On⟶GCH) |
16 | 15 | frnd 6298 | . . 3 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ran ℵ ⊆ GCH) |
17 | gch2 9832 | . . 3 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH) | |
18 | 16, 17 | sylibr 226 | . 2 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → GCH = V) |
19 | 9, 18 | impbii 201 | 1 ⊢ (GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 ⊆ wss 3791 𝒫 cpw 4378 class class class wbr 4886 ran crn 5356 Oncon0 5976 suc csuc 5978 Fn wfn 6130 ⟶wf 6131 ‘cfv 6135 ≈ cen 8238 ℵcale 9095 GCHcgch 9777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-reg 8786 ax-inf2 8835 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-seqom 7826 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-oexp 7849 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-oi 8704 df-har 8752 df-wdom 8753 df-cnf 8856 df-r1 8924 df-rank 8925 df-card 9098 df-aleph 9099 df-ac 9272 df-cda 9325 df-fin4 9444 df-gch 9778 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |