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Theorem isfin7-2 9614
Description: A set is VII-finite iff it is non-well-orderable or finite. (Contributed by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
isfin7-2 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))

Proof of Theorem isfin7-2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfin7 9519 . . . 4 (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
21ibi 259 . . 3 (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
3 isnum2 9166 . . . . 5 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 ensym 8353 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
5 simprl 759 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ On)
6 enfi 8527 . . . . . . . . . . . . . . 15 (𝐴𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7 onfin 8502 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
86, 7sylan9bbr 503 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
98biimprd 240 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝑥 ∈ ω → 𝐴 ∈ Fin))
109con3d 150 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝐴𝑥) → (¬ 𝐴 ∈ Fin → ¬ 𝑥 ∈ ω))
1110impcom 399 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → ¬ 𝑥 ∈ ω)
125, 11eldifd 3833 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ (On ∖ ω))
13 simprr 761 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝐴𝑥)
1412, 13jca 504 . . . . . . . . 9 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
154, 14sylanr2 671 . . . . . . . 8 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
1615ex 405 . . . . . . 7 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥)))
1716reximdv2 3209 . . . . . 6 𝐴 ∈ Fin → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
1817com12 32 . . . . 5 (∃𝑥 ∈ On 𝑥𝐴 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
193, 18sylbi 209 . . . 4 (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2019con1d 142 . . 3 (𝐴 ∈ dom card → (¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ Fin))
212, 20syl5com 31 . 2 (𝐴 ∈ FinVII → (𝐴 ∈ dom card → 𝐴 ∈ Fin))
22 eldifi 3986 . . . . . . 7 (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23 ensym 8353 . . . . . . 7 (𝐴𝑥𝑥𝐴)
24 isnumi 9167 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
2522, 23, 24syl2an 587 . . . . . 6 ((𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥) → 𝐴 ∈ dom card)
2625rexlimiva 3219 . . . . 5 (∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ dom card)
2726con3i 152 . . . 4 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
28 isfin7 9519 . . . 4 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2927, 28syl5ibr 238 . . 3 (𝐴𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30 fin17 9612 . . . 4 (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
3130a1i 11 . . 3 (𝐴𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
3229, 31jad 176 . 2 (𝐴𝑉 → ((𝐴 ∈ dom card → 𝐴 ∈ Fin) → 𝐴 ∈ FinVII))
3321, 32impbid2 218 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wcel 2051  wrex 3082  cdif 3819   class class class wbr 4925  dom cdm 5403  Oncon0 6026  ωcom 7394  cen 8301  Fincfn 8304  cardccrd 9156  FinVIIcfin7 9502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-om 7395  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-card 9160  df-fin7 9509
This theorem is referenced by:  fin71num  9615  dffin7-2  9616
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