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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 483 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → 𝑥 ∈ On) | |
2 | fvex 6909 | . . . . 5 ⊢ (ℵ‘𝑥) ∈ V | |
3 | simpl 481 | . . . . 5 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → GCH = V) | |
4 | 2, 3 | eleqtrrid 2832 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH) |
5 | fvex 6909 | . . . . 5 ⊢ (ℵ‘suc 𝑥) ∈ V | |
6 | 5, 3 | eleqtrrid 2832 | . . . 4 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH) |
7 | gchaleph2 10697 | . . . 4 ⊢ ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) | |
8 | 1, 4, 6, 7 | syl3anc 1368 | . . 3 ⊢ ((GCH = V ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
9 | 8 | ralrimiva 3135 | . 2 ⊢ (GCH = V → ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
10 | alephgch 10699 | . . . . . 6 ⊢ ((ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → (ℵ‘𝑥) ∈ GCH) | |
11 | 10 | ralimi 3072 | . . . . 5 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ∀𝑥 ∈ On (ℵ‘𝑥) ∈ GCH) |
12 | alephfnon 10090 | . . . . . 6 ⊢ ℵ Fn On | |
13 | ffnfv 7128 | . . . . . 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 233 | . . . 4 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ℵ:On⟶GCH) |
16 | 15 | frnd 6731 | . . 3 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → ran ℵ ⊆ GCH) |
17 | gch2 10700 | . . 3 ⊢ (GCH = V ↔ ran ℵ ⊆ GCH) | |
18 | 16, 17 | sylibr 233 | . 2 ⊢ (∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥) → GCH = V) |
19 | 9, 18 | impbii 208 | 1 ⊢ (GCH = V ↔ ∀𝑥 ∈ On (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ⊆ wss 3944 𝒫 cpw 4604 class class class wbr 5149 ran crn 5679 Oncon0 6371 suc csuc 6373 Fn wfn 6544 ⟶wf 6545 ‘cfv 6549 ≈ cen 8961 ℵcale 9961 GCHcgch 10645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-reg 9617 ax-inf2 9666 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-seqom 8469 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-oexp 8493 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-oi 9535 df-har 9582 df-wdom 9590 df-cnf 9687 df-r1 9789 df-rank 9790 df-dju 9926 df-card 9964 df-aleph 9965 df-ac 10141 df-fin4 10312 df-gch 10646 |
This theorem is referenced by: (None) |
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