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Theorem gch2 10674
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 4007 . . 3 ran ℵ ⊆ V
2 sseq2 4009 . . 3 (GCH = V → (ran ℵ ⊆ GCH ↔ ran ℵ ⊆ V))
31, 2mpbiri 257 . 2 (GCH = V → ran ℵ ⊆ GCH)
4 cardidm 9958 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
5 iscard3 10092 . . . . . . . 8 ((card‘(card‘𝑥)) = (card‘𝑥) ↔ (card‘𝑥) ∈ (ω ∪ ran ℵ))
64, 5mpbi 229 . . . . . . 7 (card‘𝑥) ∈ (ω ∪ ran ℵ)
7 elun 4149 . . . . . . 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 10622 . . . . . . . . 9 Fin ⊆ GCH
10 nnfi 9171 . . . . . . . . 9 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
119, 10sselid 3981 . . . . . . . 8 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH)
1211a1i 11 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ GCH))
13 ssel 3976 . . . . . . 7 (ran ℵ ⊆ GCH → ((card‘𝑥) ∈ ran ℵ → (card‘𝑥) ∈ GCH))
1412, 13jaod 855 . . . . . 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 3476 . . . . . . 7 𝑥 ∈ V
17 alephon 10068 . . . . . . . . . . 11 (ℵ‘suc 𝑥) ∈ On
18 simpr 483 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
19 simpl 481 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → ran ℵ ⊆ GCH)
20 alephfnon 10064 . . . . . . . . . . . . . 14 ℵ Fn On
21 fnfvelrn 7083 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2220, 18, 21sylancr 585 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ ran ℵ)
2319, 22sseldd 3984 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘𝑥) ∈ GCH)
24 onsuc 7803 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → suc 𝑥 ∈ On)
2524adantl 480 . . . . . . . . . . . . . 14 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → suc 𝑥 ∈ On)
26 fnfvelrn 7083 . . . . . . . . . . . . . 14 ((ℵ Fn On ∧ suc 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2720, 25, 26sylancr 585 . . . . . . . . . . . . 13 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ ran ℵ)
2819, 27sseldd 3984 . . . . . . . . . . . 12 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ∈ GCH)
29 gchaleph2 10671 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ (ℵ‘𝑥) ∈ GCH ∧ (ℵ‘suc 𝑥) ∈ GCH) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
3018, 23, 28, 29syl3anc 1369 . . . . . . . . . . 11 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥))
31 isnumi 9945 . . . . . . . . . . 11 (((ℵ‘suc 𝑥) ∈ On ∧ (ℵ‘suc 𝑥) ≈ 𝒫 (ℵ‘𝑥)) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3217, 30, 31sylancr 585 . . . . . . . . . 10 ((ran ℵ ⊆ GCH ∧ 𝑥 ∈ On) → 𝒫 (ℵ‘𝑥) ∈ dom card)
3332ralrimiva 3144 . . . . . . . . 9 (ran ℵ ⊆ GCH → ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
34 dfac12 10148 . . . . . . . . 9 (CHOICE ↔ ∀𝑥 ∈ On 𝒫 (ℵ‘𝑥) ∈ dom card)
3533, 34sylibr 233 . . . . . . . 8 (ran ℵ ⊆ GCH → CHOICE)
36 dfac10 10136 . . . . . . . 8 (CHOICE ↔ dom card = V)
3735, 36sylib 217 . . . . . . 7 (ran ℵ ⊆ GCH → dom card = V)
3816, 37eleqtrrid 2838 . . . . . 6 (ran ℵ ⊆ GCH → 𝑥 ∈ dom card)
39 cardid2 9952 . . . . . 6 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
40 engch 10627 . . . . . 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 2728 . 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 394  wo 843   = wceq 1539  wcel 2104  wral 3059  Vcvv 3472  cun 3947  wss 3949  𝒫 cpw 4603   class class class wbr 5149  dom cdm 5677  ran crn 5678  Oncon0 6365  suc csuc 6367   Fn wfn 6539  cfv 6544  ωcom 7859  cen 8940  Fincfn 8943  cardccrd 9934  cale 9935  CHOICEwac 10114  GCHcgch 10619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-reg 9591  ax-inf2 9640
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8151  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-seqom 8452  df-1o 8470  df-2o 8471  df-oadd 8474  df-omul 8475  df-oexp 8476  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9366  df-oi 9509  df-har 9556  df-wdom 9564  df-cnf 9661  df-r1 9763  df-rank 9764  df-dju 9900  df-card 9938  df-aleph 9939  df-ac 10115  df-fin4 10286  df-gch 10620
This theorem is referenced by:  gch3  10675
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