Theorem bj-tagss 37124

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 37119	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 37114	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5299	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5250	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4739	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 230	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4141	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3978	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2113   ∪ cun 3897   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  {csn 4578  sngl bj-csngl 37109  tag bj-ctag 37118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rex 3059  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-pw 4554  df-sn 4579  df-pr 4581  df-bj-sngl 37110  df-bj-tag 37119

This theorem is referenced by: (None)