Theorem bj-tagss 35170

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

Step	Hyp	Ref	Expression
1		df-bj-tag 35165	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 35160	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5278	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5231	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4719	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 229	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4119	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3955	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2106   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  sngl bj-csngl 35155  tag bj-ctag 35164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-pw 4535  df-sn 4562  df-pr 4564  df-bj-sngl 35156  df-bj-tag 35165

This theorem is referenced by: (None)