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Theorem isfin7-2 10436
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 10341 . . . 4 (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
21ibi 267 . . 3 (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
3 isnum2 9985 . . . . 5 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 ensym 9043 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
5 simprl 771 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ On)
6 enfi 9227 . . . . . . . . . . . . . . 15 (𝐴𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7 onfin 9267 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
86, 7sylan9bbr 510 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
98biimprd 248 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝑥 ∈ ω → 𝐴 ∈ Fin))
109con3d 152 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝐴𝑥) → (¬ 𝐴 ∈ Fin → ¬ 𝑥 ∈ ω))
1110impcom 407 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → ¬ 𝑥 ∈ ω)
125, 11eldifd 3962 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ (On ∖ ω))
13 simprr 773 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝐴𝑥)
1412, 13jca 511 . . . . . . . . 9 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
154, 14sylanr2 683 . . . . . . . 8 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
1615ex 412 . . . . . . 7 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥)))
1716reximdv2 3164 . . . . . 6 𝐴 ∈ Fin → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
1817com12 32 . . . . 5 (∃𝑥 ∈ On 𝑥𝐴 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
193, 18sylbi 217 . . . 4 (𝐴 ∈ dom card → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2019con1d 145 . . 3 (𝐴 ∈ dom card → (¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ Fin))
212, 20syl5com 31 . 2 (𝐴 ∈ FinVII → (𝐴 ∈ dom card → 𝐴 ∈ Fin))
22 eldifi 4131 . . . . . . 7 (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23 ensym 9043 . . . . . . 7 (𝐴𝑥𝑥𝐴)
24 isnumi 9986 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
2522, 23, 24syl2an 596 . . . . . 6 ((𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥) → 𝐴 ∈ dom card)
2625rexlimiva 3147 . . . . 5 (∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ dom card)
2726con3i 154 . . . 4 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
28 isfin7 10341 . . . 4 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2927, 28imbitrrid 246 . . 3 (𝐴𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30 fin17 10434 . . . 4 (𝐴 ∈ Fin → 𝐴 ∈ FinVII)
3130a1i 11 . . 3 (𝐴𝑉 → (𝐴 ∈ Fin → 𝐴 ∈ FinVII))
3229, 31jad 187 . 2 (𝐴𝑉 → ((𝐴 ∈ dom card → 𝐴 ∈ Fin) → 𝐴 ∈ FinVII))
3321, 32impbid2 226 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wcel 2108  wrex 3070  cdif 3948   class class class wbr 5143  dom cdm 5685  Oncon0 6384  ωcom 7887  cen 8982  Fincfn 8985  cardccrd 9975  FinVIIcfin7 10324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-fin7 10331
This theorem is referenced by:  fin71num  10437  dffin7-2  10438
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