Theorem bj-tagss 36975

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 36970	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 36965	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5314	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5265	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4752	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 230	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4157	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3996	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2109   ∪ cun 3915   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  sngl bj-csngl 36960  tag bj-ctag 36969

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rex 3055  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593  df-pr 4595  df-bj-sngl 36961  df-bj-tag 36970

This theorem is referenced by: (None)