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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 7629. (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 7627 | . . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
2 | df-nel 3045 | . . 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 | snex 5357 | . . 3 ⊢ {𝑦} ∈ V | |
5 | vsnid 4601 | . . 3 ⊢ 𝑦 ∈ {𝑦} | |
6 | 4, 5 | pm3.2i 470 | . 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) |
7 | 3, 6 | mpg 1795 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∀wal 1535 = wceq 1537 ∃wex 1777 ∈ wcel 2101 {cab 2710 ∉ wnel 3044 Vcvv 3434 {csn 4564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3224 df-v 3436 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-sn 4565 df-pr 4567 df-uni 4842 df-iun 4929 |
This theorem is referenced by: fiprc 8859 |
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