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Theorem bj-tagss 33918
Description: The tagging of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-tagss tag 𝐴 ⊆ 𝒫 𝐴

Proof of Theorem bj-tagss
StepHypRef Expression
1 df-bj-tag 33913 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
2 bj-snglss 33908 . . 3 sngl 𝐴 ⊆ 𝒫 𝐴
3 0elpw 5154 . . . 4 ∅ ∈ 𝒫 𝐴
4 0ex 5109 . . . . 5 ∅ ∈ V
54snss 4631 . . . 4 (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
63, 5mpbi 231 . . 3 {∅} ⊆ 𝒫 𝐴
72, 6unssi 4088 . 2 (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
81, 7eqsstri 3928 1 tag 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2083  cun 3863  wss 3865  c0 4217  𝒫 cpw 4459  {csn 4478  sngl bj-csngl 33903  tag bj-ctag 33912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-pw 4461  df-sn 4479  df-pr 4481  df-bj-sngl 33904  df-bj-tag 33913
This theorem is referenced by: (None)
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