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Theorem bj-0eltag 33291
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 4943 . . . . 5 ∅ ∈ V
21snid 4354 . . . 4 ∅ ∈ {∅}
32olci 405 . . 3 (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4 elun 3897 . . 3 (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
53, 4mpbir 221 . 2 ∅ ∈ (sngl 𝐴 ∪ {∅})
6 df-bj-tag 33288 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
75, 6eleqtrri 2839 1 ∅ ∈ tag 𝐴
Colors of variables: wff setvar class
Syntax hints:  wo 382  wcel 2140  cun 3714  c0 4059  {csn 4322  sngl bj-csngl 33278  tag bj-ctag 33287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-nul 4942
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-dif 3719  df-un 3721  df-nul 4060  df-sn 4323  df-bj-tag 33288
This theorem is referenced by:  bj-tagn0  33292
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