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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 476 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → 𝑥 ∈ On) | |
2 | fvex 6113 | . . . . 5 ⊢ (ℵ‘𝑥) ∈ V | |
3 | simpl 472 | . . . . 5 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → GCH = V) | |
4 | 2, 3 | syl5eleqr 2695 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH) |
5 | fvex 6113 | . . . . 5 ⊢ (ℵ‘suc 𝑥) ∈ V | |
6 | 5, 3 | syl5eleqr 2695 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH) |
7 | gchaleph2 9373 | . . . 4 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) | |
8 | 1, 4, 6, 7 | syl3anc 1318 | . . 3 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
9 | 8 | ralrimiva 2949 | . 2 ⊢ (GCH = V → ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
10 | alephgch 9375 | . . . . . 6 ⊢ ((ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → (ℵ‘𝑥) ∈ GCH) | |
11 | 10 | ralimi 2936 | . . . . 5 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
12 | alephfnon 8771 | . . . . . 6 ⊢ ℵ Fn On | |
13 | ffnfv 6295 | . . . . . 6 ⊢ (ℵ:On⟶GCH ↔ (ℵ Fn On ∧ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH)) | |
14 | 12, 13 | mpbiran 955 | . . . . 5 ⊢ (ℵ:On⟶GCH ↔ ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
15 | 11, 14 | sylibr 223 | . . . 4 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ℵ:On⟶GCH) |
16 | df-f 5808 | . . . . 5 ⊢ (ℵ:On⟶GCH ↔ (ℵ Fn On ∧ ran ℵ ⊆ GCH)) | |
17 | 12, 16 | mpbiran 955 | . . . 4 ⊢ (ℵ:On⟶GCH ↔ ran ℵ ⊆ GCH) |
18 | 15, 17 | sylib 207 | . . 3 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ran ℵ ⊆ GCH) |
19 | gch2 9376 | . . 3 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH) | |
20 | 18, 19 | sylibr 223 | . 2 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → GCH = V) |
21 | 9, 20 | impbii 198 | 1 ⊢ (GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 ran crn 5039 Oncon0 5640 suc csuc 5642 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 ≈ cen 7838 ℵcale 8645 GCHcgch 9321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-reg 8380 ax-inf2 8421 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-har 8346 df-wdom 8347 df-cnf 8442 df-r1 8510 df-rank 8511 df-card 8648 df-aleph 8649 df-ac 8822 df-cda 8873 df-fin4 8992 df-gch 9322 |
This theorem is referenced by: (None) |
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