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Mirrors > Home > MPE Home > Th. List > ruALT | Structured version Visualization version GIF version |
Description: Alternate proof of ru 3715, simplified using (indirectly) the Axiom of Regularity ax-reg 9351. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ruALT | ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 5239 | . . 3 ⊢ ¬ V ∈ V | |
2 | 1 | nelir 3052 | . 2 ⊢ V ∉ V |
3 | ruv 9361 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
4 | neleq1 3054 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} = V → ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V)) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V ↔ V ∉ V) |
6 | 2, 5 | mpbir 230 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 {cab 2715 ∉ wnel 3049 Vcvv 3432 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-reg 9351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nel 3050 df-ral 3069 df-rex 3070 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-sn 4562 df-pr 4564 |
This theorem is referenced by: (None) |
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