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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ru1 | Structured version Visualization version GIF version |
Description: A version of Russell's paradox ru 3401 (see also bj-ru 32126). Note the more economical use of bj-abeq2 31961 instead of abeq2 2719 to avoid dependency on ax-13 2234. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ru1 | ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-ru0 32124 | . . 3 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) | |
2 | bj-abeq2 31961 | . . 3 ⊢ (𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} ↔ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥)) | |
3 | 1, 2 | mtbir 312 | . 2 ⊢ ¬ 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} |
4 | 3 | nex 1722 | 1 ⊢ ¬ ∃𝑦 𝑦 = {𝑥 ∣ ¬ 𝑥 ∈ 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 = wceq 1475 ∃wex 1695 {cab 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: bj-ru 32126 |
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