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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ru0 | Structured version Visualization version GIF version |
Description: The FOL part of Russell's paradox ru 3434 (see also bj-ru1 32933, bj-ru 32934). Use of elequ1 1997, bj-elequ12 32668, bj-spvv 32723 (instead of eleq1 2689, eleq12d 2695, spv 2260 as in ru 3434) permits to remove dependency on ax-10 2019, ax-11 2034, ax-12 2047, ax-13 2246, ax-ext 2602, df-sb 1881, df-clab 2609, df-cleq 2615, df-clel 2618. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-ru0 | ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm5.19 375 | . 2 ⊢ ¬ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦) | |
2 | elequ1 1997 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦)) | |
3 | bj-elequ12 32668 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) | |
4 | 3 | anidms 677 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦)) |
5 | 4 | notbid 308 | . . . 4 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦)) |
6 | 2, 5 | bibi12d 335 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) ↔ (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦))) |
7 | 6 | bj-spvv 32723 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) → (𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦)) |
8 | 1, 7 | mto 188 | 1 ⊢ ¬ ∀𝑥(𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: bj-ru1 32933 |
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