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Theorem bj-0eltag 32159
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 4718 . . . . 5 ∅ ∈ V
21snid 4155 . . . 4 ∅ ∈ {∅}
32olci 405 . . 3 (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅})
4 elun 3715 . . 3 (∅ ∈ (sngl 𝐴 ∪ {∅}) ↔ (∅ ∈ sngl 𝐴 ∨ ∅ ∈ {∅}))
53, 4mpbir 220 . 2 ∅ ∈ (sngl 𝐴 ∪ {∅})
6 df-bj-tag 32156 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
75, 6eleqtrri 2687 1 ∅ ∈ tag 𝐴
Colors of variables: wff setvar class
Syntax hints:  wo 382  wcel 1977  cun 3538  c0 3874  {csn 4125  sngl bj-csngl 32146  tag bj-ctag 32155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-bj-tag 32156
This theorem is referenced by:  bj-tagn0  32160
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