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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 7246 | . 2 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | |
2 | snfi 8328 | . . . . . . . 8 ⊢ {𝑦} ∈ Fin | |
3 | eleq1 2847 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑥 ∈ Fin ↔ {𝑦} ∈ Fin)) | |
4 | 2, 3 | mpbiri 250 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → 𝑥 ∈ Fin) |
5 | 4 | exlimiv 1973 | . . . . . 6 ⊢ (∃𝑦 𝑥 = {𝑦} → 𝑥 ∈ Fin) |
6 | 5 | abssi 3898 | . . . . 5 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin |
7 | ssexg 5043 | . . . . 5 ⊢ (({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ⊆ Fin ∧ Fin ∈ V) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
8 | 6, 7 | mpan 680 | . . . 4 ⊢ (Fin ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) |
9 | 8 | con3i 152 | . . 3 ⊢ (¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ¬ Fin ∈ V) |
10 | df-nel 3076 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
11 | df-nel 3076 | . . 3 ⊢ (Fin ∉ V ↔ ¬ Fin ∈ V) | |
12 | 9, 10, 11 | 3imtr4i 284 | . 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 1601 ∃wex 1823 ∈ wcel 2107 {cab 2763 ∉ wnel 3075 Vcvv 3398 ⊆ wss 3792 {csn 4398 Fincfn 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-om 7346 df-1o 7845 df-en 8244 df-fin 8247 |
This theorem is referenced by: (None) |
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