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Theorem gchina 9372
Description: Assuming the GCH, weakly and strongly inaccessible cardinals coincide. Theorem 11.20 of [TakeutiZaring] p. 106. (Contributed by Mario Carneiro, 5-Jun-2015.)
Assertion
Ref Expression
gchina (GCH = V → Inaccw = Inacc)

Proof of Theorem gchina
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 475 . . . . 5 ((GCH = V ∧ 𝑥 ∈ Inaccw) → 𝑥 ∈ Inaccw)
2 idd 24 . . . . . . 7 ((GCH = V ∧ 𝑥 ∈ Inaccw) → (𝑥 ≠ ∅ → 𝑥 ≠ ∅))
3 idd 24 . . . . . . 7 ((GCH = V ∧ 𝑥 ∈ Inaccw) → ((cf‘𝑥) = 𝑥 → (cf‘𝑥) = 𝑥))
4 pwfi 8116 . . . . . . . . . . . . 13 (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
5 isfinite 8404 . . . . . . . . . . . . . 14 (𝒫 𝑦 ∈ Fin ↔ 𝒫 𝑦 ≺ ω)
6 winainf 9367 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Inaccw → ω ⊆ 𝑥)
7 ssdomg 7859 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Inaccw → (ω ⊆ 𝑥 → ω ≼ 𝑥))
86, 7mpd 15 . . . . . . . . . . . . . . 15 (𝑥 ∈ Inaccw → ω ≼ 𝑥)
9 sdomdomtr 7950 . . . . . . . . . . . . . . . 16 ((𝒫 𝑦 ≺ ω ∧ ω ≼ 𝑥) → 𝒫 𝑦𝑥)
109expcom 449 . . . . . . . . . . . . . . 15 (ω ≼ 𝑥 → (𝒫 𝑦 ≺ ω → 𝒫 𝑦𝑥))
118, 10syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ Inaccw → (𝒫 𝑦 ≺ ω → 𝒫 𝑦𝑥))
125, 11syl5bi 230 . . . . . . . . . . . . 13 (𝑥 ∈ Inaccw → (𝒫 𝑦 ∈ Fin → 𝒫 𝑦𝑥))
134, 12syl5bi 230 . . . . . . . . . . . 12 (𝑥 ∈ Inaccw → (𝑦 ∈ Fin → 𝒫 𝑦𝑥))
1413ad3antlr 762 . . . . . . . . . . 11 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝑦 ∈ Fin → 𝒫 𝑦𝑥))
1514a1dd 47 . . . . . . . . . 10 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝑦 ∈ Fin → (𝑦𝑧 → 𝒫 𝑦𝑥)))
16 vex 3170 . . . . . . . . . . . . . . 15 𝑦 ∈ V
17 simplll 793 . . . . . . . . . . . . . . 15 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → GCH = V)
1816, 17syl5eleqr 2689 . . . . . . . . . . . . . 14 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → 𝑦 ∈ GCH)
19 simprr 791 . . . . . . . . . . . . . 14 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → ¬ 𝑦 ∈ Fin)
20 gchinf 9330 . . . . . . . . . . . . . 14 ((𝑦 ∈ GCH ∧ ¬ 𝑦 ∈ Fin) → ω ≼ 𝑦)
2118, 19, 20syl2anc 690 . . . . . . . . . . . . 13 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → ω ≼ 𝑦)
22 vex 3170 . . . . . . . . . . . . . 14 𝑧 ∈ V
2322, 17syl5eleqr 2689 . . . . . . . . . . . . 13 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → 𝑧 ∈ GCH)
24 gchpwdom 9343 . . . . . . . . . . . . 13 ((ω ≼ 𝑦𝑦 ∈ GCH ∧ 𝑧 ∈ GCH) → (𝑦𝑧 ↔ 𝒫 𝑦𝑧))
2521, 18, 23, 24syl3anc 1317 . . . . . . . . . . . 12 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → (𝑦𝑧 ↔ 𝒫 𝑦𝑧))
26 winacard 9365 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Inaccw → (card‘𝑥) = 𝑥)
27 iscard 8656 . . . . . . . . . . . . . . . . . 18 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑧𝑥 𝑧𝑥))
2827simprbi 478 . . . . . . . . . . . . . . . . 17 ((card‘𝑥) = 𝑥 → ∀𝑧𝑥 𝑧𝑥)
2926, 28syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Inaccw → ∀𝑧𝑥 𝑧𝑥)
3029ad2antlr 758 . . . . . . . . . . . . . . 15 (((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) → ∀𝑧𝑥 𝑧𝑥)
3130r19.21bi 2910 . . . . . . . . . . . . . 14 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → 𝑧𝑥)
32 domsdomtr 7952 . . . . . . . . . . . . . . 15 ((𝒫 𝑦𝑧𝑧𝑥) → 𝒫 𝑦𝑥)
3332expcom 449 . . . . . . . . . . . . . 14 (𝑧𝑥 → (𝒫 𝑦𝑧 → 𝒫 𝑦𝑥))
3431, 33syl 17 . . . . . . . . . . . . 13 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝒫 𝑦𝑧 → 𝒫 𝑦𝑥))
3534adantrr 748 . . . . . . . . . . . 12 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → (𝒫 𝑦𝑧 → 𝒫 𝑦𝑥))
3625, 35sylbid 228 . . . . . . . . . . 11 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → (𝑦𝑧 → 𝒫 𝑦𝑥))
3736expr 640 . . . . . . . . . 10 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (¬ 𝑦 ∈ Fin → (𝑦𝑧 → 𝒫 𝑦𝑥)))
3815, 37pm2.61d 168 . . . . . . . . 9 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝑦𝑧 → 𝒫 𝑦𝑥))
3938rexlimdva 3007 . . . . . . . 8 (((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) → (∃𝑧𝑥 𝑦𝑧 → 𝒫 𝑦𝑥))
4039ralimdva 2939 . . . . . . 7 ((GCH = V ∧ 𝑥 ∈ Inaccw) → (∀𝑦𝑥𝑧𝑥 𝑦𝑧 → ∀𝑦𝑥 𝒫 𝑦𝑥))
412, 3, 403anim123d 1397 . . . . . 6 ((GCH = V ∧ 𝑥 ∈ Inaccw) → ((𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 𝑦𝑧) → (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥 𝒫 𝑦𝑥)))
42 elwina 9359 . . . . . 6 (𝑥 ∈ Inaccw ↔ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 𝑦𝑧))
43 elina 9360 . . . . . 6 (𝑥 ∈ Inacc ↔ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥 𝒫 𝑦𝑥))
4441, 42, 433imtr4g 283 . . . . 5 ((GCH = V ∧ 𝑥 ∈ Inaccw) → (𝑥 ∈ Inaccw𝑥 ∈ Inacc))
451, 44mpd 15 . . . 4 ((GCH = V ∧ 𝑥 ∈ Inaccw) → 𝑥 ∈ Inacc)
4645ex 448 . . 3 (GCH = V → (𝑥 ∈ Inaccw𝑥 ∈ Inacc))
47 inawina 9363 . . 3 (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw)
4846, 47impbid1 213 . 2 (GCH = V → (𝑥 ∈ Inaccw𝑥 ∈ Inacc))
4948eqrdv 2602 1 (GCH = V → Inaccw = Inacc)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2774  wral 2890  wrex 2891  Vcvv 3167  wss 3534  c0 3868  𝒫 cpw 4102   class class class wbr 4572  Oncon0 5621  cfv 5785  ωcom 6929  cdom 7811  csdm 7812  Fincfn 7813  cardccrd 8616  cfccf 8618  GCHcgch 9293  Inaccwcwina 9355  Inacccina 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-rep 4688  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-inf2 8393
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rmo 2898  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-int 4400  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-se 4983  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-isom 5794  df-riota 6484  df-ov 6525  df-oprab 6526  df-mpt2 6527  df-om 6930  df-1st 7031  df-2nd 7032  df-supp 7155  df-wrecs 7266  df-recs 7327  df-rdg 7365  df-seqom 7402  df-1o 7419  df-2o 7420  df-oadd 7423  df-omul 7424  df-oexp 7425  df-er 7601  df-map 7718  df-en 7814  df-dom 7815  df-sdom 7816  df-fin 7817  df-fsupp 8131  df-oi 8270  df-har 8318  df-wdom 8319  df-cnf 8414  df-card 8620  df-cf 8622  df-cda 8845  df-fin4 8964  df-gch 9294  df-wina 9357  df-ina 9358
This theorem is referenced by: (None)
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