Theorem fiprc 8993

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 7713	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2		snfi 8992	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
3		eleq1 2825	. . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
4	2, 3	mpbiri 258	. . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
5	4	exlimiv 1932	. . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
6	5	abssi 4022	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
7		ssexg 5270	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
8	6, 7	mpan 691	. . . 4 ⊢ (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
9	8	con3i 154	. . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
10		df-nel 3038	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
11		df-nel 3038	. . 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 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ∉ wnel 3037  Vcvv 3442   ⊆ wss 3903  {csn 4582  Fincfn 8895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-om 7819  df-1o 8407  df-en 8896  df-fin 8899

This theorem is referenced by: (None)