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Mirrors > Home > MPE Home > Th. List > ruALT | Structured version Visualization version GIF version |
Description: Alternate proof of ru 3777, simplified using (indirectly) the Axiom of Regularity ax-reg 9623. (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 5319 | . . 3 ⊢ ¬ V ∈ V | |
2 | 1 | nelir 3046 | . 2 ⊢ V ∉ V |
3 | ruv 9633 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
4 | neleq1 3049 | . . 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 1533 {cab 2705 ∉ wnel 3043 Vcvv 3473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-pr 5433 ax-reg 9623 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nel 3044 df-ral 3059 df-rex 3068 df-v 3475 df-un 3954 df-sn 4633 df-pr 4635 |
This theorem is referenced by: (None) |
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