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Theorem gch2 9535
 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 3658 . . 3 ran ℵ ⊆ V
2 sseq2 3660 . . 3 (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
31, 2mpbiri 248 . 2 (GCH = V → ran ℵ ⊆ GCH)
4 cardidm 8823 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
5 iscard3 8954 . . . . . . . 8 ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
64, 5mpbi 220 . . . . . . 7 (card‘𝑥) ∈ (ω ∪ ran ℵ)
7 elun 3786 . . . . . . 7 ((card‘𝑥) ∈ (ω ∪ ran ℵ) ↔ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ))
86, 7mpbi 220 . . . . . 6 ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9 fingch 9483 . . . . . . . . 9 Fin ⊆ GCH
10 nnfi 8194 . . . . . . . . 9 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
119, 10sseldi 3634 . . . . . . . 8 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
1211a1i 11 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13 ssel 3630 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
1412, 13jaod 394 . . . . . 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 3234 . . . . . . 7 𝑥 ∈ V
17 alephon 8930 . . . . . . . . . . 11 (ℵ‘suc 𝑥) ∈ On
18 simpr 476 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
19 simpl 472 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → ran ℵ ⊆ GCH)
20 alephfnon 8926 . . . . . . . . . . . . . 14 ℵ Fn On
21 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2220, 18, 21sylancr 696 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2319, 22sseldd 3637 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24 suceloni 7055 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → suc 𝑥 ∈ On)
2524adantl 481 . . . . . . . . . . . . . 14 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2720, 25, 26sylancr 696 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2819, 27sseldd 3637 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29 gchaleph2 9532 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
3018, 23, 28, 29syl3anc 1366 . . . . . . . . . . 11 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31 isnumi 8810 . . . . . . . . . . 11 (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3217, 30, 31sylancr 696 . . . . . . . . . 10 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3332ralrimiva 2995 . . . . . . . . 9 (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34 dfac12 9009 . . . . . . . . 9 (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
3533, 34sylibr 224 . . . . . . . 8 (ran ℵ ⊆ GCH → CHOICE)
36 dfac10 8997 . . . . . . . 8 (CHOICE ↔ dom card = V)
3735, 36sylib 208 . . . . . . 7 (ran ℵ ⊆ GCH → dom card = V)
3816, 37syl5eleqr 2737 . . . . . 6 (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39 cardid2 8817 . . . . . 6 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40 engch 9488 . . . . . 6 ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4138, 39, 403syl 18 . . . . 5 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4215, 41mpbid 222 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
4316a1i 11 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ V)
4442, 432thd 255 . . 3 (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
4544eqrdv 2649 . 2 (ran ℵ ⊆ GCH → GCH = V)
463, 45impbii 199 1 (GCH = V ↔ ran ℵ ⊆ GCH)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  𝒫 cpw 4191   class class class wbr 4685  dom cdm 5143  ran crn 5144  Oncon0 5761  suc csuc 5763   Fn wfn 5921  ‘cfv 5926  ωcom 7107   ≈ cen 7994  Fincfn 7997  cardccrd 8799  ℵcale 8800  CHOICEwac 8976  GCHcgch 9480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-har 8504  df-wdom 8505  df-cnf 8597  df-r1 8665  df-rank 8666  df-card 8803  df-aleph 8804  df-ac 8977  df-cda 9028  df-fin4 9147  df-gch 9481 This theorem is referenced by:  gch3  9536
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