Theorem fiprc 8977

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.

Step	Hyp	Ref	Expression
1		snnex 7700	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2		snfi 8976	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
3		eleq1 2821	. . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
4	2, 3	mpbiri 258	. . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
5	4	exlimiv 1931	. . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
6	5	abssi 4017	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
7		ssexg 5265	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
8	6, 7	mpan 690	. . . 4 ⊢ (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
9	8	con3i 154	. . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
10		df-nel 3034	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
11		df-nel 3034	. . 3 ⊢ (Fin ∉ V ↔ ¬ Fin ∈ V)
12	9, 10, 11	3imtr4i 292	. 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V → Fin ∉ V)
13	1, 12	ax-mp 5	1 ⊢ Fin ∉ V

Syntax hints:  ¬ wn 3   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711   ∉ wnel 3033  Vcvv 3437   ⊆ wss 3898  {csn 4577  Fincfn 8879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-om 7806  df-1o 8394  df-en 8880  df-fin 8883

This theorem is referenced by: (None)