ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fiprc GIF version

Theorem fiprc 6841
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 4466 . 2 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2 vex 2755 . . . . . . . . 9 𝑦 ∈ V
3 snfig 6840 . . . . . . . . 9 (𝑦 ∈ V → {𝑦} ∈ Fin)
42, 3ax-mp 5 . . . . . . . 8 {𝑦} ∈ Fin
5 eleq1 2252 . . . . . . . 8 (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
64, 5mpbiri 168 . . . . . . 7 (𝑥 = {𝑦} → 𝑥 ∈ Fin)
76exlimiv 1609 . . . . . 6 (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
87abssi 3245 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
9 ssexg 4157 . . . . 5 (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
108, 9mpan 424 . . . 4 (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
1110con3i 633 . . 3 (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
12 df-nel 2456 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
13 df-nel 2456 . . 3 (Fin ∉ V ↔ ¬ Fin ∈ V)
1411, 12, 133imtr4i 201 . 2 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V → Fin ∉ V)
151, 14ax-mp 5 1 Fin ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1364  wex 1503  wcel 2160  {cab 2175  wnel 2455  Vcvv 2752  wss 3144  {csn 3607  Fincfn 6766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-nel 2456  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-id 4311  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-1o 6441  df-en 6767  df-fin 6769
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator