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| Description: Alternate proof of ru 3786, simplified using (indirectly) the Axiom of Regularity ax-reg 9632. (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 5315 | . . 3 ⊢ ¬ V ∈ V | |
| 2 | 1 | nelir 3049 | . 2 ⊢ V ∉ V | 
| 3 | ruv 9642 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
| 4 | neleq1 3052 | . . 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 231 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 {cab 2714 ∉ wnel 3046 Vcvv 3480 | 
| 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-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-pr 5432 ax-reg 9632 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nel 3047 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 | 
| This theorem is referenced by: (None) | 
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