Theorem bj-tagss 35651

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 35646	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 35641	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5346	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5299	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4781	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 229	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4180	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 4011	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2106   ∪ cun 3941   ⊆ wss 3943  ∅c0 4317  𝒫 cpw 4595  {csn 4621  sngl bj-csngl 35636  tag bj-ctag 35645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-pw 4597  df-sn 4622  df-pr 4624  df-bj-sngl 35637  df-bj-tag 35646

This theorem is referenced by: (None)