Theorem fiprc 6985

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 4543	. 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
2		vex 2803	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		snfig 6984	. . . . . . . . 9 ⊢ (𝑦 ∈ V → {𝑦} ∈ Fin)
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ {𝑦} ∈ Fin
5		eleq1 2292	. . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin))
6	4, 5	mpbiri 168	. . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin)
7	6	exlimiv 1644	. . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin)
8	7	abssi 3300	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin
9		ssexg 4226	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
10	8, 9	mpan 424	. . . 4 ⊢ (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
11	10	con3i 635	. . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V)
12		df-nel 2496	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
13		df-nel 2496	. . 3 ⊢ (Fin ∉ V ↔ ¬ Fin ∈ V)
14	11, 12, 13	3imtr4i 201	. 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V → Fin ∉ V)
15	1, 14	ax-mp 5	1 ⊢ Fin ∉ V

Syntax hints:  ¬ wn 3   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215   ∉ wnel 2495  Vcvv 2800   ⊆ wss 3198  {csn 3667  Fincfn 6904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-nel 2496  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-1o 6577  df-en 6905  df-fin 6907

This theorem is referenced by: (None)