Theorem bj-tagss 36164

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 36159	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 36154	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5353	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5306	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4788	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 229	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4184	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 4015	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2104   ∪ cun 3945   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627  sngl bj-csngl 36149  tag bj-ctag 36158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rex 3069  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-pw 4603  df-sn 4628  df-pr 4630  df-bj-sngl 36150  df-bj-tag 36159

This theorem is referenced by: (None)