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Theorem isfin7-2 10393
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 10298 . . . 4 (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
21ibi 267 . . 3 (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
3 isnum2 9942 . . . . 5 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 ensym 9001 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
5 simprl 768 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ On)
6 enfi 9192 . . . . . . . . . . . . . . 15 (𝐴𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7 onfin 9232 . . . . . . . . . . . . . . 15 (𝑥 ∈ On → (𝑥 ∈ Fin ↔ 𝑥 ∈ ω))
86, 7sylan9bbr 510 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝐴𝑥) → (𝐴 ∈ Fin ↔ 𝑥 ∈ ω))
98biimprd 247 . . . . . . . . . . . . 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 3954 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ (On ∖ ω))
13 simprr 770 . . . . . . . . . 10 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝐴𝑥)
1412, 13jca 511 . . . . . . . . 9 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
154, 14sylanr2 680 . . . . . . . 8 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝑥𝐴)) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥))
1615ex 412 . . . . . . 7 𝐴 ∈ Fin → ((𝑥 ∈ On ∧ 𝑥𝐴) → (𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥)))
1716reximdv2 3158 . . . . . 6 𝐴 ∈ Fin → (∃𝑥 ∈ On 𝑥𝐴 → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
1817com12 32 . . . . 5 (∃𝑥 ∈ On 𝑥𝐴 → (¬ 𝐴 ∈ Fin → ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
193, 18sylbi 216 . . . 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 4121 . . . . . . 7 (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23 ensym 9001 . . . . . . 7 (𝐴𝑥𝑥𝐴)
24 isnumi 9943 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
2522, 23, 24syl2an 595 . . . . . 6 ((𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥) → 𝐴 ∈ dom card)
2625rexlimiva 3141 . . . . 5 (∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ dom card)
2726con3i 154 . . . 4 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
28 isfin7 10298 . . . 4 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2927, 28imbitrrid 245 . . 3 (𝐴𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30 fin17 10391 . . . 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 225 1 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ (𝐴 ∈ dom card → 𝐴 ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wcel 2098  wrex 3064  cdif 3940   class class class wbr 5141  dom cdm 5669  Oncon0 6358  ωcom 7852  cen 8938  Fincfn 8941  cardccrd 9932  FinVIIcfin7 10281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-fin7 10288
This theorem is referenced by:  fin71num  10394  dffin7-2  10395
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