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Theorem bj-0eltag 36979
Description: The empty set belongs to the tagging of a class. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-0eltag ∅ ∈ tag 𝐴

Proof of Theorem bj-0eltag
StepHypRef Expression
1 0ex 5307 . . . . 5 ∅ ∈ V
21snid 4662 . . . 4 ∅ ∈ {∅}
32olci 867 . . 3 (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4 elun 4153 . . 3 (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
53, 4mpbir 231 . 2 ∅ ∈ (sngl 𝐴 ∪ {∅})
6 df-bj-tag 36976 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
75, 6eleqtrri 2840 1 ∅ ∈ tag 𝐴
Colors of variables: wff setvar class
Syntax hints:  wo 848  wcel 2108  cun 3949  c0 4333  {csn 4626  sngl bj-csngl 36966  tag bj-ctag 36975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-nul 4334  df-sn 4627  df-bj-tag 36976
This theorem is referenced by:  bj-tagn0  36980
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