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Theorem isfin7-2 9811
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 9716 . . . 4 (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
21ibi 270 . . 3 (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
3 isnum2 9362 . . . . 5 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 ensym 8545 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
5 simprl 770 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ On)
6 enfi 8722 . . . . . . . . . . . . . . 15 (𝐴𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7 onfin 8698 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
86, 7sylan9bbr 514 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
98biimprd 251 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝑥 ∈ ω → 𝐴 ∈ Fin))
109con3d 155 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝐴𝑥) → (¬ 𝐴 ∈ Fin → ¬ 𝑥 ∈ ω))
1110impcom 411 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → ¬ 𝑥 ∈ ω)
125, 11eldifd 3895 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ (On ∖ ω))
13 simprr 772 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝐴𝑥)
1412, 13jca 515 . . . . . . . . 9 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
154, 14sylanr2 682 . . . . . . . 8 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
1615ex 416 . . . . . . 7 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥)))
1716reximdv2 3233 . . . . . 6 𝐴 ∈ Fin → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
1817com12 32 . . . . 5 (∃𝑥 ∈ On 𝑥𝐴 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
193, 18sylbi 220 . . . 4 (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2019con1d 147 . . 3 (𝐴 ∈ dom card → (¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ Fin))
212, 20syl5com 31 . 2 (𝐴 ∈ FinVII → (𝐴 ∈ dom card → 𝐴 ∈ Fin))
22 eldifi 4057 . . . . . . 7 (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23 ensym 8545 . . . . . . 7 (𝐴𝑥𝑥𝐴)
24 isnumi 9363 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
2522, 23, 24syl2an 598 . . . . . 6 ((𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥) → 𝐴 ∈ dom card)
2625rexlimiva 3243 . . . . 5 (∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ dom card)
2726con3i 157 . . . 4 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
28 isfin7 9716 . . . 4 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2927, 28syl5ibr 249 . . 3 (𝐴𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30 fin17 9809 . . . 4 (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
3130a1i 11 . . 3 (𝐴𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
3229, 31jad 190 . 2 (𝐴𝑉 → ((𝐴 ∈ dom card → 𝐴 ∈ Fin) → 𝐴 ∈ FinVII))
3321, 32impbid2 229 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2112  wrex 3110  cdif 3881   class class class wbr 5033  dom cdm 5523  Oncon0 6163  ωcom 7564  cen 8493  Fincfn 8496  cardccrd 9352  FinVIIcfin7 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-om 7565  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-fin7 9706
This theorem is referenced by:  fin71num  9812  dffin7-2  9813
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