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| Description: The class of all singletons is a proper class. See also pwnex 7779. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.) | 
| Ref | Expression | 
|---|---|
| snnex | ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | abnex 7777 | . . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
| 2 | df-nel 3047 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
| 3 | 1, 2 | sylibr 234 | . 2 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V) | 
| 4 | vsnex 5434 | . . 3 ⊢ {𝑦} ∈ V | |
| 5 | vsnid 4663 | . . 3 ⊢ 𝑦 ∈ {𝑦} | |
| 6 | 4, 5 | pm3.2i 470 | . 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) | 
| 7 | 3, 6 | mpg 1797 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∉ wnel 3046 Vcvv 3480 {csn 4626 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-un 3956 df-in 3958 df-ss 3968 df-sn 4627 df-pr 4629 df-uni 4908 df-iun 4993 | 
| This theorem is referenced by: fiprc 9085 | 
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