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Theorem bj-0eltag 33396
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 4952 . . . . 5 ∅ ∈ V
21snid 4368 . . . 4 ∅ ∈ {∅}
32olci 892 . . 3 (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4 elun 3917 . . 3 (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
53, 4mpbir 222 . 2 ∅ ∈ (sngl 𝐴 ∪ {∅})
6 df-bj-tag 33393 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
75, 6eleqtrri 2843 1 ∅ ∈ tag 𝐴
Colors of variables: wff setvar class
Syntax hints:  wo 873  wcel 2155  cun 3732  c0 4081  {csn 4336  sngl bj-csngl 33383  tag bj-ctag 33392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4951
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3737  df-un 3739  df-nul 4082  df-sn 4337  df-bj-tag 33393
This theorem is referenced by:  bj-tagn0  33397
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