![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fiprc | Structured version Visualization version GIF version |
Description: The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.) |
Ref | Expression |
---|---|
fiprc | ⊢ Fin ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snnex 7697 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | |
2 | snfi 8995 | . . . . . . . 8 ⊢ {𝑦} ∈ Fin | |
3 | eleq1 2826 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin)) | |
4 | 2, 3 | mpbiri 258 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin) |
5 | 4 | exlimiv 1934 | . . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin) |
6 | 5 | abssi 4032 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin |
7 | ssexg 5285 | . . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
8 | 6, 7 | mpan 689 | . . . 4 ⊢ (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) |
9 | 8 | con3i 154 | . . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V) |
10 | df-nel 3051 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
11 | df-nel 3051 | . . 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 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2714 ∉ wnel 3050 Vcvv 3448 ⊆ wss 3915 {csn 4591 Fincfn 8890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-om 7808 df-1o 8417 df-en 8891 df-fin 8894 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |