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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 7563. (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 7561 | . . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
2 | df-nel 3048 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V) | |
3 | 1, 2 | sylibr 237 | . 2 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V) |
4 | snex 5339 | . . 3 ⊢ {𝑦} ∈ V | |
5 | vsnid 4593 | . . 3 ⊢ 𝑦 ∈ {𝑦} | |
6 | 4, 5 | pm3.2i 474 | . 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) |
7 | 3, 6 | mpg 1805 | 1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2111 {cab 2715 ∉ wnel 3047 Vcvv 3421 {csn 4556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-nel 3048 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-sn 4557 df-pr 4559 df-uni 4835 df-iun 4921 |
This theorem is referenced by: fiprc 8745 |
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