Theorem bj-tagss 36587

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 36582	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 36577	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5356	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5308	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4791	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 229	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4183	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 4011	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2098   ∪ cun 3942   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604  {csn 4630  sngl bj-csngl 36572  tag bj-ctag 36581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rex 3060  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-pw 4606  df-sn 4631  df-pr 4633  df-bj-sngl 36573  df-bj-tag 36582

This theorem is referenced by: (None)