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Theorem fiprc 8996
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 7697 . 2 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2 snfi 8995 . . . . . . . 8 {𝑦} ∈ Fin
3 eleq1 2826 . . . . . . . 8 (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
42, 3mpbiri 258 . . . . . . 7 (𝑥 = {𝑦} → 𝑥 ∈ Fin)
54exlimiv 1934 . . . . . 6 (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
65abssi 4032 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
7 ssexg 5285 . . . . 5 (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
86, 7mpan 689 . . . 4 (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
98con3i 154 . . 3 (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
10 df-nel 3051 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
11 df-nel 3051 . . 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 1542  wex 1782  wcel 2107  {cab 2714  wnel 3050  Vcvv 3448  wss 3915  {csn 4591  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-om 7808  df-1o 8417  df-en 8891  df-fin 8894
This theorem is referenced by: (None)
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