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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 7761 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | |
2 | snfi 9069 | . . . . . . . 8 ⊢ {𝑦} ∈ Fin | |
3 | eleq1 2813 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin)) | |
4 | 2, 3 | mpbiri 257 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin) |
5 | 4 | exlimiv 1925 | . . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin) |
6 | 5 | abssi 4063 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin |
7 | ssexg 5324 | . . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
8 | 6, 7 | mpan 688 | . . . 4 ⊢ (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) |
9 | 8 | con3i 154 | . . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V) |
10 | df-nel 3036 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
11 | df-nel 3036 | . . 3 ⊢ (Fin ∉ V ↔ ¬ Fin ∈ V) | |
12 | 9, 10, 11 | 3imtr4i 291 | . 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 1533 ∃wex 1773 ∈ wcel 2098 {cab 2702 ∉ wnel 3035 Vcvv 3461 ⊆ wss 3944 {csn 4630 Fincfn 8964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-om 7872 df-1o 8487 df-en 8965 df-fin 8968 |
This theorem is referenced by: (None) |
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