Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-snglc Structured version   Visualization version   Unicode version

Theorem bj-snglc 31633
 Description: Characterization of the elements of in terms of elements of its singletonization. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglc sngl

Proof of Theorem bj-snglc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-rex 2762 . 2
2 bj-elsngl 31632 . 2 sngl
3 elisset 3043 . . . . 5
43pm4.71i 644 . . . 4
5 19.42v 1842 . . . 4
6 eleq1 2537 . . . . . . 7
76eqcoms 2479 . . . . . 6
87pm5.32ri 650 . . . . 5
98exbii 1726 . . . 4
104, 5, 93bitr2i 281 . . 3
11 vex 3034 . . . . . . 7
12 sneqbg 4134 . . . . . . 7
1311, 12ax-mp 5 . . . . . 6
14 eqcom 2478 . . . . . 6
1513, 14bitr3i 259 . . . . 5
1615anbi2i 708 . . . 4
1716exbii 1726 . . 3
1810, 17bitri 257 . 2
191, 2, 183bitr4ri 286 1 sngl
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376   wceq 1452  wex 1671   wcel 1904  wrex 2757  cvv 3031  csn 3959  sngl bj-csngl 31629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962  df-bj-sngl 31630 This theorem is referenced by:  bj-snglinv  31636  bj-tagci  31648  bj-tagcg  31649
 Copyright terms: Public domain W3C validator