![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fiprc | 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 4466 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | |
2 | vex 2755 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
3 | snfig 6840 | . . . . . . . . 9 ⊢ (𝑦 ∈ V → {𝑦} ∈ Fin) | |
4 | 2, 3 | ax-mp 5 | . . . . . . . 8 ⊢ {𝑦} ∈ Fin |
5 | eleq1 2252 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin)) | |
6 | 4, 5 | mpbiri 168 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin) |
7 | 6 | exlimiv 1609 | . . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin) |
8 | 7 | abssi 3245 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin |
9 | ssexg 4157 | . . . . 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 633 | . . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V) |
12 | df-nel 2456 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
13 | df-nel 2456 | . . 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 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1364 ∃wex 1503 ∈ wcel 2160 {cab 2175 ∉ wnel 2455 Vcvv 2752 ⊆ wss 3144 {csn 3607 Fincfn 6766 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-nel 2456 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-1o 6441 df-en 6767 df-fin 6769 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |