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Theorem hargch 9480
Description: If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 9486. (Contributed by Mario Carneiro, 2-Jun-2015.)
Assertion
Ref Expression
hargch ((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ GCH)

Proof of Theorem hargch
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 harcl 8451 . . . . . . . . . . . . . 14 (har‘𝐴) ∈ On
2 sdomdom 7968 . . . . . . . . . . . . . 14 (𝑥 ≺ (har‘𝐴) → 𝑥 ≼ (har‘𝐴))
3 ondomen 8845 . . . . . . . . . . . . . 14 (((har‘𝐴) ∈ On ∧ 𝑥 ≼ (har‘𝐴)) → 𝑥 ∈ dom card)
41, 2, 3sylancr 694 . . . . . . . . . . . . 13 (𝑥 ≺ (har‘𝐴) → 𝑥 ∈ dom card)
5 onenon 8760 . . . . . . . . . . . . . 14 ((har‘𝐴) ∈ On → (har‘𝐴) ∈ dom card)
61, 5ax-mp 5 . . . . . . . . . . . . 13 (har‘𝐴) ∈ dom card
7 cardsdom2 8799 . . . . . . . . . . . . 13 ((𝑥 ∈ dom card ∧ (har‘𝐴) ∈ dom card) → ((card‘𝑥) ∈ (card‘(har‘𝐴)) ↔ 𝑥 ≺ (har‘𝐴)))
84, 6, 7sylancl 693 . . . . . . . . . . . 12 (𝑥 ≺ (har‘𝐴) → ((card‘𝑥) ∈ (card‘(har‘𝐴)) ↔ 𝑥 ≺ (har‘𝐴)))
98ibir 257 . . . . . . . . . . 11 (𝑥 ≺ (har‘𝐴) → (card‘𝑥) ∈ (card‘(har‘𝐴)))
10 harcard 8789 . . . . . . . . . . 11 (card‘(har‘𝐴)) = (har‘𝐴)
119, 10syl6eleq 2709 . . . . . . . . . 10 (𝑥 ≺ (har‘𝐴) → (card‘𝑥) ∈ (har‘𝐴))
12 elharval 8453 . . . . . . . . . . 11 ((card‘𝑥) ∈ (har‘𝐴) ↔ ((card‘𝑥) ∈ On ∧ (card‘𝑥) ≼ 𝐴))
1312simprbi 480 . . . . . . . . . 10 ((card‘𝑥) ∈ (har‘𝐴) → (card‘𝑥) ≼ 𝐴)
1411, 13syl 17 . . . . . . . . 9 (𝑥 ≺ (har‘𝐴) → (card‘𝑥) ≼ 𝐴)
15 cardid2 8764 . . . . . . . . . 10 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
16 domen1 8087 . . . . . . . . . 10 ((card‘𝑥) ≈ 𝑥 → ((card‘𝑥) ≼ 𝐴𝑥𝐴))
174, 15, 163syl 18 . . . . . . . . 9 (𝑥 ≺ (har‘𝐴) → ((card‘𝑥) ≼ 𝐴𝑥𝐴))
1814, 17mpbid 222 . . . . . . . 8 (𝑥 ≺ (har‘𝐴) → 𝑥𝐴)
19 domnsym 8071 . . . . . . . 8 (𝑥𝐴 → ¬ 𝐴𝑥)
2018, 19syl 17 . . . . . . 7 (𝑥 ≺ (har‘𝐴) → ¬ 𝐴𝑥)
2120con2i 134 . . . . . 6 (𝐴𝑥 → ¬ 𝑥 ≺ (har‘𝐴))
22 sdomen2 8090 . . . . . . 7 ((har‘𝐴) ≈ 𝒫 𝐴 → (𝑥 ≺ (har‘𝐴) ↔ 𝑥 ≺ 𝒫 𝐴))
2322notbid 308 . . . . . 6 ((har‘𝐴) ≈ 𝒫 𝐴 → (¬ 𝑥 ≺ (har‘𝐴) ↔ ¬ 𝑥 ≺ 𝒫 𝐴))
2421, 23syl5ib 234 . . . . 5 ((har‘𝐴) ≈ 𝒫 𝐴 → (𝐴𝑥 → ¬ 𝑥 ≺ 𝒫 𝐴))
25 imnan 438 . . . . 5 ((𝐴𝑥 → ¬ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
2624, 25sylib 208 . . . 4 ((har‘𝐴) ≈ 𝒫 𝐴 → ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
2726alrimiv 1853 . . 3 ((har‘𝐴) ≈ 𝒫 𝐴 → ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
2827olcd 408 . 2 ((har‘𝐴) ≈ 𝒫 𝐴 → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
29 relen 7945 . . . . 5 Rel ≈
3029brrelex2i 5149 . . . 4 ((har‘𝐴) ≈ 𝒫 𝐴 → 𝒫 𝐴 ∈ V)
31 pwexb 6960 . . . 4 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
3230, 31sylibr 224 . . 3 ((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ V)
33 elgch 9429 . . 3 (𝐴 ∈ V → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
3432, 33syl 17 . 2 ((har‘𝐴) ≈ 𝒫 𝐴 → (𝐴 ∈ GCH ↔ (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))))
3528, 34mpbird 247 1 ((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ GCH)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  wal 1479  wcel 1988  Vcvv 3195  𝒫 cpw 4149   class class class wbr 4644  dom cdm 5104  Oncon0 5711  cfv 5876  cen 7937  cdom 7938  csdm 7939  Fincfn 7940  harchar 8446  cardccrd 8746  GCHcgch 9427
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
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  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-wrecs 7392  df-recs 7453  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-oi 8400  df-har 8448  df-card 8750  df-gch 9428
This theorem is referenced by: (None)
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