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Theorem gch2 10669
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 4006 . . 3 ran ℵ ⊆ V
2 sseq2 4008 . . 3 (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
31, 2mpbiri 257 . 2 (GCH = V → ran ℵ ⊆ GCH)
4 cardidm 9953 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
5 iscard3 10087 . . . . . . . 8 ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
64, 5mpbi 229 . . . . . . 7 (card‘𝑥) ∈ (ω ∪ ran ℵ)
7 elun 4148 . . . . . . 7 ((card‘𝑥) ∈ (ω ∪ ran ℵ) ↔ ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ))
86, 7mpbi 229 . . . . . 6 ((card‘𝑥) ∈ ω ∨ (card‘𝑥) ∈ ran ℵ)
9 fingch 10617 . . . . . . . . 9 Fin ⊆ GCH
10 nnfi 9166 . . . . . . . . 9 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
119, 10sselid 3980 . . . . . . . 8 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
1211a1i 11 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13 ssel 3975 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
1412, 13jaod 857 . . . . . 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 3478 . . . . . . 7 𝑥 ∈ V
17 alephon 10063 . . . . . . . . . . 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 10059 . . . . . . . . . . . . . 14 ℵ Fn On
21 fnfvelrn 7082 . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24 onsuc 7798 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → suc 𝑥 ∈ On)
2524adantl 482 . . . . . . . . . . . . . 14 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26 fnfvelrn 7082 . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29 gchaleph2 10666 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
3018, 23, 28, 29syl3anc 1371 . . . . . . . . . . 11 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31 isnumi 9940 . . . . . . . . . . 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 3146 . . . . . . . . 9 (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34 dfac12 10143 . . . . . . . . 9 (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
3533, 34sylibr 233 . . . . . . . 8 (ran ℵ ⊆ GCH → CHOICE)
36 dfac10 10131 . . . . . . . 8 (CHOICE ↔ dom card = V)
3735, 36sylib 217 . . . . . . 7 (ran ℵ ⊆ GCH → dom card = V)
3816, 37eleqtrrid 2840 . . . . . 6 (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39 cardid2 9947 . . . . . 6 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40 engch 10622 . . . . . 6 ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4138, 39, 403syl 18 . . . . 5 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ GCH ↔ 𝑥 ∈ GCH))
4215, 41mpbid 231 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ GCH)
4316a1i 11 . . . 4 (ran ℵ ⊆ GCH → 𝑥 ∈ V)
4442, 432thd 264 . . 3 (ran ℵ ⊆ GCH → (𝑥 ∈ GCH ↔ 𝑥 ∈ V))
4544eqrdv 2730 . 2 (ran ℵ ⊆ GCH → GCH = V)
463, 45impbii 208 1 (GCH = V ↔ ran ℵ ⊆ GCH)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3061  Vcvv 3474  cun 3946  wss 3948  𝒫 cpw 4602   class class class wbr 5148  dom cdm 5676  ran crn 5677  Oncon0 6364  suc csuc 6366   Fn wfn 6538  cfv 6543  ωcom 7854  cen 8935  Fincfn 8938  cardccrd 9929  cale 9930  CHOICEwac 10109  GCHcgch 10614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-reg 9586  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seqom 8447  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470  df-oexp 8471  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-oi 9504  df-har 9551  df-wdom 9559  df-cnf 9656  df-r1 9758  df-rank 9759  df-dju 9895  df-card 9933  df-aleph 9934  df-ac 10110  df-fin4 10281  df-gch 10615
This theorem is referenced by:  gch3  10670
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