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Theorem gchina 9506
 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 477 . . . . 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 8246 . . . . . . . . . . . . 13 (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
5 isfinite 8534 . . . . . . . . . . . . . 14 (𝒫 𝑦 ∈ Fin ↔ 𝒫 𝑦 ≺ ω)
6 winainf 9501 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Inaccw → ω ⊆ 𝑥)
7 ssdomg 7986 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Inaccw → (ω ⊆ 𝑥 → ω ≼ 𝑥))
86, 7mpd 15 . . . . . . . . . . . . . . 15 (𝑥 ∈ Inaccw → ω ≼ 𝑥)
9 sdomdomtr 8078 . . . . . . . . . . . . . . . 16 ((𝒫 𝑦 ≺ ω ∧ ω ≼ 𝑥) → 𝒫 𝑦𝑥)
109expcom 451 . . . . . . . . . . . . . . 15 (ω ≼ 𝑥 → (𝒫 𝑦 ≺ ω → 𝒫 𝑦𝑥))
118, 10syl 17 . . . . . . . . . . . . . 14 (𝑥 ∈ Inaccw → (𝒫 𝑦 ≺ ω → 𝒫 𝑦𝑥))
125, 11syl5bi 232 . . . . . . . . . . . . 13 (𝑥 ∈ Inaccw → (𝒫 𝑦 ∈ Fin → 𝒫 𝑦𝑥))
134, 12syl5bi 232 . . . . . . . . . . . 12 (𝑥 ∈ Inaccw → (𝑦 ∈ Fin → 𝒫 𝑦𝑥))
1413ad3antlr 766 . . . . . . . . . . 11 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝑦 ∈ Fin → 𝒫 𝑦𝑥))
1514a1dd 50 . . . . . . . . . 10 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝑦 ∈ Fin → (𝑦𝑧 → 𝒫 𝑦𝑥)))
16 vex 3198 . . . . . . . . . . . . . . 15 𝑦 ∈ V
17 simplll 797 . . . . . . . . . . . . . . 15 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → GCH = V)
1816, 17syl5eleqr 2706 . . . . . . . . . . . . . 14 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → 𝑦 ∈ GCH)
19 simprr 795 . . . . . . . . . . . . . 14 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → ¬ 𝑦 ∈ Fin)
20 gchinf 9464 . . . . . . . . . . . . . 14 ((𝑦 ∈ GCH ∧ ¬ 𝑦 ∈ Fin) → ω ≼ 𝑦)
2118, 19, 20syl2anc 692 . . . . . . . . . . . . 13 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → ω ≼ 𝑦)
22 vex 3198 . . . . . . . . . . . . . 14 𝑧 ∈ V
2322, 17syl5eleqr 2706 . . . . . . . . . . . . 13 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → 𝑧 ∈ GCH)
24 gchpwdom 9477 . . . . . . . . . . . . 13 ((ω ≼ 𝑦𝑦 ∈ GCH ∧ 𝑧 ∈ GCH) → (𝑦𝑧 ↔ 𝒫 𝑦𝑧))
2521, 18, 23, 24syl3anc 1324 . . . . . . . . . . . 12 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → (𝑦𝑧 ↔ 𝒫 𝑦𝑧))
26 winacard 9499 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Inaccw → (card‘𝑥) = 𝑥)
27 iscard 8786 . . . . . . . . . . . . . . . . . 18 ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑧𝑥 𝑧𝑥))
2827simprbi 480 . . . . . . . . . . . . . . . . 17 ((card‘𝑥) = 𝑥 → ∀𝑧𝑥 𝑧𝑥)
2926, 28syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Inaccw → ∀𝑧𝑥 𝑧𝑥)
3029ad2antlr 762 . . . . . . . . . . . . . . 15 (((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) → ∀𝑧𝑥 𝑧𝑥)
3130r19.21bi 2929 . . . . . . . . . . . . . 14 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → 𝑧𝑥)
32 domsdomtr 8080 . . . . . . . . . . . . . . 15 ((𝒫 𝑦𝑧𝑧𝑥) → 𝒫 𝑦𝑥)
3332expcom 451 . . . . . . . . . . . . . 14 (𝑧𝑥 → (𝒫 𝑦𝑧 → 𝒫 𝑦𝑥))
3431, 33syl 17 . . . . . . . . . . . . 13 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝒫 𝑦𝑧 → 𝒫 𝑦𝑥))
3534adantrr 752 . . . . . . . . . . . 12 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → (𝒫 𝑦𝑧 → 𝒫 𝑦𝑥))
3625, 35sylbid 230 . . . . . . . . . . 11 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ (𝑧𝑥 ∧ ¬ 𝑦 ∈ Fin)) → (𝑦𝑧 → 𝒫 𝑦𝑥))
3736expr 642 . . . . . . . . . 10 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (¬ 𝑦 ∈ Fin → (𝑦𝑧 → 𝒫 𝑦𝑥)))
3815, 37pm2.61d 170 . . . . . . . . 9 ((((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) ∧ 𝑧𝑥) → (𝑦𝑧 → 𝒫 𝑦𝑥))
3938rexlimdva 3027 . . . . . . . 8 (((GCH = V ∧ 𝑥 ∈ Inaccw) ∧ 𝑦𝑥) → (∃𝑧𝑥 𝑦𝑧 → 𝒫 𝑦𝑥))
4039ralimdva 2959 . . . . . . 7 ((GCH = V ∧ 𝑥 ∈ Inaccw) → (∀𝑦𝑥𝑧𝑥 𝑦𝑧 → ∀𝑦𝑥 𝒫 𝑦𝑥))
412, 3, 403anim123d 1404 . . . . . 6 ((GCH = V ∧ 𝑥 ∈ Inaccw) → ((𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 𝑦𝑧) → (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥 𝒫 𝑦𝑥)))
42 elwina 9493 . . . . . 6 (𝑥 ∈ Inaccw ↔ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥𝑧𝑥 𝑦𝑧))
43 elina 9494 . . . . . 6 (𝑥 ∈ Inacc ↔ (𝑥 ≠ ∅ ∧ (cf‘𝑥) = 𝑥 ∧ ∀𝑦𝑥 𝒫 𝑦𝑥))
4441, 42, 433imtr4g 285 . . . . 5 ((GCH = V ∧ 𝑥 ∈ Inaccw) → (𝑥 ∈ Inaccw𝑥 ∈ Inacc))
451, 44mpd 15 . . . 4 ((GCH = V ∧ 𝑥 ∈ Inaccw) → 𝑥 ∈ Inacc)
4645ex 450 . . 3 (GCH = V → (𝑥 ∈ Inaccw𝑥 ∈ Inacc))
47 inawina 9497 . . 3 (𝑥 ∈ Inacc → 𝑥 ∈ Inaccw)
4846, 47impbid1 215 . 2 (GCH = V → (𝑥 ∈ Inaccw𝑥 ∈ Inacc))
4948eqrdv 2618 1 (GCH = V → Inaccw = Inacc)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ≠ wne 2791  ∀wral 2909  ∃wrex 2910  Vcvv 3195   ⊆ wss 3567  ∅c0 3907  𝒫 cpw 4149   class class class wbr 4644  Oncon0 5711  ‘cfv 5876  ωcom 7050   ≼ cdom 7938   ≺ csdm 7939  Fincfn 7940  cardccrd 8746  cfccf 8748  GCHcgch 9427  Inaccwcwina 9489  Inacccina 9490 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550  df-oexp 7551  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-har 8448  df-wdom 8449  df-cnf 8544  df-card 8750  df-cf 8752  df-cda 8975  df-fin4 9094  df-gch 9428  df-wina 9491  df-ina 9492 This theorem is referenced by: (None)
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