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Theorem fiprc 9085
Description: The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.)
Assertion
Ref Expression
fiprc Fin ∉ V

Proof of Theorem fiprc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snnex 7778 . 2 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2 snfi 9083 . . . . . . . 8 {𝑦} ∈ Fin
3 eleq1 2829 . . . . . . . 8 (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
42, 3mpbiri 258 . . . . . . 7 (𝑥 = {𝑦} → 𝑥 ∈ Fin)
54exlimiv 1930 . . . . . 6 (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
65abssi 4070 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
7 ssexg 5323 . . . . 5 (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
86, 7mpan 690 . . . 4 (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
98con3i 154 . . 3 (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
10 df-nel 3047 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
11 df-nel 3047 . . 3 (Fin ∉ V ↔ ¬ Fin ∈ V)
129, 10, 113imtr4i 292 . 2 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V → Fin ∉ V)
131, 12ax-mp 5 1 Fin ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wnel 3046  Vcvv 3480  wss 3951  {csn 4626  Fincfn 8985
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-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  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-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  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-iun 4993  df-br 5144  df-opab 5206  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-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by: (None)
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