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Mirrors > Home > MPE Home > Th. List > ruALT | Structured version Visualization version GIF version |
Description: Alternate proof of ru 3680, simplified using (indirectly) the Axiom of Regularity ax-reg 9132. (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 5184 | . . 3 ⊢ ¬ V ∈ V | |
2 | 1 | nelir 3042 | . 2 ⊢ V ∉ V |
3 | ruv 9142 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V | |
4 | neleq1 3044 | . . 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 234 | 1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} ∉ V |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1542 {cab 2717 ∉ wnel 3039 Vcvv 3399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-reg 9132 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-nel 3040 df-ral 3059 df-rex 3060 df-v 3401 df-dif 3847 df-un 3849 df-nul 4213 df-sn 4518 df-pr 4520 |
This theorem is referenced by: (None) |
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