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Theorem bj-ru0 31581
Description: The FOL part of Russell's paradox ru 3277 (see also bj-ru1 31582, bj-ru 31583). Use of elequ1 1904, bj-elequ12 31321, bj-spvv 31368 (instead of eleq1 2527, eleq12d 2533, spv 2114 as in ru 3277) permits to remove dependency on ax-10 1925, ax-11 1930, ax-12 1943, ax-13 2101, ax-ext 2441, df-sb 1808, df-clab 2448, df-cleq 2454, df-clel 2457. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ru0  |-  -.  A. x ( x  e.  y  <->  -.  x  e.  x )
Distinct variable group:    x, y

Proof of Theorem bj-ru0
StepHypRef Expression
1 pm5.19 366 . 2  |-  -.  (
y  e.  y  <->  -.  y  e.  y )
2 elequ1 1904 . . . 4  |-  ( x  =  y  ->  (
x  e.  y  <->  y  e.  y ) )
3 bj-elequ12 31321 . . . . . 6  |-  ( ( x  =  y  /\  x  =  y )  ->  ( x  e.  x  <->  y  e.  y ) )
43anidms 655 . . . . 5  |-  ( x  =  y  ->  (
x  e.  x  <->  y  e.  y ) )
54notbid 300 . . . 4  |-  ( x  =  y  ->  ( -.  x  e.  x  <->  -.  y  e.  y ) )
62, 5bibi12d 327 . . 3  |-  ( x  =  y  ->  (
( x  e.  y  <->  -.  x  e.  x
)  <->  ( y  e.  y  <->  -.  y  e.  y ) ) )
76bj-spvv 31368 . 2  |-  ( A. x ( x  e.  y  <->  -.  x  e.  x )  ->  (
y  e.  y  <->  -.  y  e.  y ) )
81, 7mto 181 1  |-  -.  A. x ( x  e.  y  <->  -.  x  e.  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189   A.wal 1452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674
This theorem is referenced by:  bj-ru1  31582
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