Theorem bj-tagss 35097

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 35092	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 35087	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5273	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5226	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4716	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 229	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4115	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3951	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2108   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  sngl bj-csngl 35082  tag bj-ctag 35091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rex 3069  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561  df-bj-sngl 35083  df-bj-tag 35092

This theorem is referenced by: (None)