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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 7776 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | |
2 | snfi 9081 | . . . . . . . 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 1927 | . . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin) |
6 | 5 | abssi 4079 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin |
7 | ssexg 5328 | . . . . 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 3044 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
11 | df-nel 3044 | . . 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 1536 ∃wex 1775 ∈ wcel 2105 {cab 2711 ∉ wnel 3043 Vcvv 3477 ⊆ wss 3962 {csn 4630 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-om 7887 df-1o 8504 df-en 8984 df-fin 8987 |
This theorem is referenced by: (None) |
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