Theorem bj-tagss 37342

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 37337	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 37332	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5285	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5230	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4717	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 231	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4121	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3961	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2119   ∪ cun 3881   ⊆ wss 3883  ∅c0 4262  𝒫 cpw 4530  {csn 4556  sngl bj-csngl 37327  tag bj-ctag 37336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rex 3064  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4263  df-pw 4532  df-sn 4557  df-pr 4559  df-bj-sngl 37328  df-bj-tag 37337

This theorem is referenced by: (None)