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Theorem isfin7-2 10434
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 10339 . . . 4 (𝐴 ∈ FinVII → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
21ibi 267 . . 3 (𝐴 ∈ FinVII → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
3 isnum2 9983 . . . . 5 (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥𝐴)
4 ensym 9042 . . . . . . . . 9 (𝑥𝐴𝐴𝑥)
5 simprl 771 . . . . . . . . . . 11 ((¬ 𝐴 ∈ Fin ∧ (𝑥 ∈ On ∧ 𝐴𝑥)) → 𝑥 ∈ On)
6 enfi 9225 . . . . . . . . . . . . . . 15 (𝐴𝑥 → (𝐴 ∈ Fin ↔ 𝑥 ∈ Fin))
7 onfin 9265 . . . . . . . . . . . . . . 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 3974 . . . . . . . . . 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 3162 . . . . . 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 4141 . . . . . . 7 (𝑥 ∈ (On ∖ ω) → 𝑥 ∈ On)
23 ensym 9042 . . . . . . 7 (𝐴𝑥𝑥𝐴)
24 isnumi 9984 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑥𝐴) → 𝐴 ∈ dom card)
2522, 23, 24syl2an 596 . . . . . 6 ((𝑥 ∈ (On ∖ ω) ∧ 𝐴𝑥) → 𝐴 ∈ dom card)
2625rexlimiva 3145 . . . . 5 (∃𝑥 ∈ (On ∖ ω)𝐴𝑥𝐴 ∈ dom card)
2726con3i 154 . . . 4 𝐴 ∈ dom card → ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥)
28 isfin7 10339 . . . 4 (𝐴𝑉 → (𝐴 ∈ FinVII ↔ ¬ ∃𝑥 ∈ (On ∖ ω)𝐴𝑥))
2927, 28imbitrrid 246 . . 3 (𝐴𝑉 → (¬ 𝐴 ∈ dom card → 𝐴 ∈ FinVII))
30 fin17 10432 . . . 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 2106  wrex 3068  cdif 3960   class class class wbr 5148  dom cdm 5689  Oncon0 6386  ωcom 7887  cen 8981  Fincfn 8984  cardccrd 9973  FinVIIcfin7 10322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-fin7 10329
This theorem is referenced by:  fin71num  10435  dffin7-2  10436
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