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Theorem bj-ru0 33057
Description: The FOL part of Russell's paradox ru 3467 (see also bj-ru1 33058, bj-ru 33059). Use of elequ1 2037, bj-elequ12 32793, bj-spvv 32848 (instead of eleq1 2718, eleq12d 2724, spv 2296 as in ru 3467) permits to remove dependency on ax-10 2059, ax-11 2074, ax-12 2087, ax-13 2282, ax-ext 2631, df-sb 1938, df-clab 2638, df-cleq 2644, df-clel 2647. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ru0 ¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-ru0
StepHypRef Expression
1 pm5.19 374 . 2 ¬ (𝑦𝑦 ↔ ¬ 𝑦𝑦)
2 elequ1 2037 . . . 4 (𝑥 = 𝑦 → (𝑥𝑦𝑦𝑦))
3 bj-elequ12 32793 . . . . . 6 ((𝑥 = 𝑦𝑥 = 𝑦) → (𝑥𝑥𝑦𝑦))
43anidms 678 . . . . 5 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
54notbid 307 . . . 4 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
62, 5bibi12d 334 . . 3 (𝑥 = 𝑦 → ((𝑥𝑦 ↔ ¬ 𝑥𝑥) ↔ (𝑦𝑦 ↔ ¬ 𝑦𝑦)))
76bj-spvv 32848 . 2 (∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥) → (𝑦𝑦 ↔ ¬ 𝑦𝑦))
81, 7mto 188 1 ¬ ∀𝑥(𝑥𝑦 ↔ ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  bj-ru1  33058
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