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Theorem hargch 9697
Description: If 𝐴 + ≈ 𝒫 𝐴, then 𝐴 is a GCH-set. The much simpler converse to gchhar 9703. (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 8622 . . . . . . . . . . . . . 14 (har‘𝐴) ∈ On
2 sdomdom 8137 . . . . . . . . . . . . . 14 (𝑥 ≺ (har‘𝐴) → 𝑥 ≼ (har‘𝐴))
3 ondomen 9060 . . . . . . . . . . . . . 14 (((har‘𝐴) ∈ On ∧ 𝑥 ≼ (har‘𝐴)) → 𝑥 ∈ dom card)
41, 2, 3sylancr 575 . . . . . . . . . . . . 13 (𝑥 ≺ (har‘𝐴) → 𝑥 ∈ dom card)
5 onenon 8975 . . . . . . . . . . . . . 14 ((har‘𝐴) ∈ On → (har‘𝐴) ∈ dom card)
61, 5ax-mp 5 . . . . . . . . . . . . 13 (har‘𝐴) ∈ dom card
7 cardsdom2 9014 . . . . . . . . . . . . 13 ((𝑥 ∈ dom card ∧ (har‘𝐴) ∈ dom card) → ((card‘𝑥) ∈ (card‘(har‘𝐴)) ↔ 𝑥 ≺ (har‘𝐴)))
84, 6, 7sylancl 574 . . . . . . . . . . . 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 9004 . . . . . . . . . . 11 (card‘(har‘𝐴)) = (har‘𝐴)
119, 10syl6eleq 2860 . . . . . . . . . 10 (𝑥 ≺ (har‘𝐴) → (card‘𝑥) ∈ (har‘𝐴))
12 elharval 8624 . . . . . . . . . . 11 ((card‘𝑥) ∈ (har‘𝐴) ↔ ((card‘𝑥) ∈ On ∧ (card‘𝑥) ≼ 𝐴))
1312simprbi 484 . . . . . . . . . 10 ((card‘𝑥) ∈ (har‘𝐴) → (card‘𝑥) ≼ 𝐴)
1411, 13syl 17 . . . . . . . . 9 (𝑥 ≺ (har‘𝐴) → (card‘𝑥) ≼ 𝐴)
15 cardid2 8979 . . . . . . . . . 10 (𝑥 ∈ dom card → (card‘𝑥) ≈ 𝑥)
16 domen1 8258 . . . . . . . . . 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 8242 . . . . . . . 8 (𝑥𝐴 → ¬ 𝐴𝑥)
2018, 19syl 17 . . . . . . 7 (𝑥 ≺ (har‘𝐴) → ¬ 𝐴𝑥)
2120con2i 136 . . . . . 6 (𝐴𝑥 → ¬ 𝑥 ≺ (har‘𝐴))
22 sdomen2 8261 . . . . . . 7 ((har‘𝐴) ≈ 𝒫 𝐴 → (𝑥 ≺ (har‘𝐴) ↔ 𝑥 ≺ 𝒫 𝐴))
2322notbid 307 . . . . . 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 386 . . . . 5 ((𝐴𝑥 → ¬ 𝑥 ≺ 𝒫 𝐴) ↔ ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
2624, 25sylib 208 . . . 4 ((har‘𝐴) ≈ 𝒫 𝐴 → ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
2726alrimiv 2007 . . 3 ((har‘𝐴) ≈ 𝒫 𝐴 → ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴))
2827olcd 863 . 2 ((har‘𝐴) ≈ 𝒫 𝐴 → (𝐴 ∈ Fin ∨ ∀𝑥 ¬ (𝐴𝑥𝑥 ≺ 𝒫 𝐴)))
29 relen 8114 . . . . 5 Rel ≈
3029brrelex2i 5299 . . . 4 ((har‘𝐴) ≈ 𝒫 𝐴 → 𝒫 𝐴 ∈ V)
31 pwexb 7122 . . . 4 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
3230, 31sylibr 224 . . 3 ((har‘𝐴) ≈ 𝒫 𝐴𝐴 ∈ V)
33 elgch 9646 . . 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  wa 382  wo 836  wal 1629  wcel 2145  Vcvv 3351  𝒫 cpw 4297   class class class wbr 4786  dom cdm 5249  Oncon0 5866  cfv 6031  cen 8106  cdom 8107  csdm 8108  Fincfn 8109  harchar 8617  cardccrd 8961  GCHcgch 9644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-wrecs 7559  df-recs 7621  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-oi 8571  df-har 8619  df-card 8965  df-gch 9645
This theorem is referenced by: (None)
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