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Mirrors > Home > MPE Home > Th. List > snnex | Structured version Visualization version GIF version |
Description: The class of all singletons is a proper class. See also pwnex 7698. (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 7696 | . . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
2 | df-nel 3051 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
3 | 1, 2 | sylibr 233 | . 2 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V) |
4 | vsnex 5391 | . . 3 ⊢ {𝑦} ∈ V | |
5 | vsnid 4628 | . . 3 ⊢ 𝑦 ∈ {𝑦} | |
6 | 4, 5 | pm3.2i 472 | . 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) |
7 | 3, 6 | mpg 1800 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2714 ∉ wnel 3050 Vcvv 3448 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2708 ax-sep 5261 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-un 3920 df-in 3922 df-ss 3932 df-sn 4592 df-pr 4594 df-uni 4871 df-iun 4961 |
This theorem is referenced by: fiprc 8996 |
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