MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gch2 Structured version   Visualization version   GIF version

Theorem gch2 10086
Description: It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gch2 (GCH = V ↔ ran ℵ ⊆ GCH)

Proof of Theorem gch2
StepHypRef Expression
1 ssv 3990 . . 3 ran ℵ ⊆ V
2 sseq2 3992 . . 3 (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
31, 2mpbiri 259 . 2 (GCH = V → ran ℵ ⊆ GCH)
4 cardidm 9377 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
5 iscard3 9508 . . . . . . . 8 ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
64, 5mpbi 231 . . . . . . 7 (card‘𝑥) ∈ (ω ∪ ran ℵ)
7 elun 4124 . . . . . . 7 ((card‘𝑥) ∈ (ω ∪ ran ℵ) ↔ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ))
86, 7mpbi 231 . . . . . 6 ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9 fingch 10034 . . . . . . . . 9 Fin ⊆ GCH
10 nnfi 8700 . . . . . . . . 9 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
119, 10sseldi 3964 . . . . . . . 8 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
1211a1i 11 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13 ssel 3960 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
1412, 13jaod 853 . . . . . 6 (ran ℵ ⊆ GCH → (((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ) → (card‘𝑥) ∈ GCH))
158, 14mpi 20 . . . . 5 (ran ℵ ⊆ GCH → (card‘𝑥) ∈ GCH)
16 vex 3498 . . . . . . 7 𝑥 ∈ V
17 alephon 9484 . . . . . . . . . . 11 (ℵ‘suc 𝑥) ∈ On
18 simpr 485 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
19 simpl 483 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → ran ℵ ⊆ GCH)
20 alephfnon 9480 . . . . . . . . . . . . . 14 ℵ Fn On
21 fnfvelrn 6841 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2220, 18, 21sylancr 587 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2319, 22sseldd 3967 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24 suceloni 7516 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → suc 𝑥 ∈ On)
2524adantl 482 . . . . . . . . . . . . . 14 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26 fnfvelrn 6841 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2720, 25, 26sylancr 587 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2819, 27sseldd 3967 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29 gchaleph2 10083 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
3018, 23, 28, 29syl3anc 1363 . . . . . . . . . . 11 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31 isnumi 9364 . . . . . . . . . . 11 (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3217, 30, 31sylancr 587 . . . . . . . . . 10 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3332ralrimiva 3182 . . . . . . . . 9 (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34 dfac12 9564 . . . . . . . . 9 (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
3533, 34sylibr 235 . . . . . . . 8 (ran ℵ ⊆ GCH → CHOICE)
36 dfac10 9552 . . . . . . . 8 (CHOICE ↔ dom card = V)
3735, 36sylib 219 . . . . . . 7 (ran ℵ ⊆ GCH → dom card = V)
3816, 37eleqtrrid 2920 . . . . . 6 (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39 cardid2 9371 . . . . . 6 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40 engch 10039 . . . . . 6 ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4138, 39, 403syl 18 . . . . 5 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4215, 41mpbid 233 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
4316a1i 11 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ V)
4442, 432thd 266 . . 3 (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
4544eqrdv 2819 . 2 (ran ℵ ⊆ GCH → GCH = V)
463, 45impbii 210 1 (GCH = V ↔ ran ℵ ⊆ GCH)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wral 3138  Vcvv 3495  cun 3933  wss 3935  𝒫 cpw 4537   class class class wbr 5058  dom cdm 5549  ran crn 5550  Oncon0 6185  suc csuc 6187   Fn wfn 6344  cfv 6349  ωcom 7568  cen 8495  Fincfn 8498  cardccrd 9353  cale 9354  CHOICEwac 9530  GCHcgch 10031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-reg 9045  ax-inf2 9093
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  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 3497  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 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-seqom 8075  df-1o 8093  df-2o 8094  df-oadd 8097  df-omul 8098  df-oexp 8099  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-oi 8963  df-har 9011  df-wdom 9012  df-cnf 9114  df-r1 9182  df-rank 9183  df-dju 9319  df-card 9357  df-aleph 9358  df-ac 9531  df-fin4 9698  df-gch 10032
This theorem is referenced by:  gch3  10087
  Copyright terms: Public domain W3C validator