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Theorem bj-tagss 33299
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 33294 . 2 tag 𝐴 = (sngl 𝐴 ∪ {∅})
2 bj-snglss 33289 . . 3 sngl 𝐴 ⊆ 𝒫 𝐴
3 0elpw 4965 . . . 4 ∅ ∈ 𝒫 𝐴
4 0ex 4924 . . . . 5 ∅ ∈ V
54snss 4451 . . . 4 (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
63, 5mpbi 220 . . 3 {∅} ⊆ 𝒫 𝐴
72, 6unssi 3939 . 2 (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
81, 7eqsstri 3784 1 tag 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  cun 3721  wss 3723  c0 4063  𝒫 cpw 4297  {csn 4316  sngl bj-csngl 33284  tag bj-ctag 33293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-pw 4299  df-sn 4317  df-pr 4319  df-bj-sngl 33285  df-bj-tag 33294
This theorem is referenced by: (None)
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