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Theorem bj-tagss 33280
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 33275 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
2 bj-snglss 33270 . . 3 sngl 𝐴 ⊆ 𝒫 𝐴
3 0elpw 5033 . . . 4 ∅ ∈ 𝒫 𝐴
4 0ex 4991 . . . . 5 ∅ ∈ V
54snss 4513 . . . 4 (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
63, 5mpbi 221 . . 3 {∅} ⊆ 𝒫 𝐴
72, 6unssi 3994 . 2 (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
81, 7eqsstri 3839 1 tag 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  cun 3774  wss 3776  c0 4123  𝒫 cpw 4358  {csn 4377  sngl bj-csngl 33265  tag bj-ctag 33274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-pw 4360  df-sn 4378  df-pr 4380  df-bj-sngl 33266  df-bj-tag 33275
This theorem is referenced by: (None)
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