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Theorem fiprc 6512
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 4262 . 2 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2 vex 2622 . . . . . . . . 9 𝑦 ∈ V
3 snfig 6511 . . . . . . . . 9 (𝑦 ∈ V → {𝑦} ∈ Fin)
42, 3ax-mp 7 . . . . . . . 8 {𝑦} ∈ Fin
5 eleq1 2150 . . . . . . . 8 (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
64, 5mpbiri 166 . . . . . . 7 (𝑥 = {𝑦} → 𝑥 ∈ Fin)
76exlimiv 1534 . . . . . 6 (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
87abssi 3094 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
9 ssexg 3970 . . . . 5 (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
108, 9mpan 415 . . . 4 (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
1110con3i 597 . . 3 (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
12 df-nel 2351 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
13 df-nel 2351 . . 3 (Fin ∉ V ↔ ¬ Fin ∈ V)
1411, 12, 133imtr4i 199 . 2 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V → Fin ∉ V)
151, 14ax-mp 7 1 Fin ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1289  wex 1426  wcel 1438  {cab 2074  wnel 2350  Vcvv 2619  wss 2997  {csn 3441  Fincfn 6437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-nel 2351  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-1o 6163  df-en 6438  df-fin 6440
This theorem is referenced by: (None)
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