Theorem bj-tagss 37024

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 37019	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 37014	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5292	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5243	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4734	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 230	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4138	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3976	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2111   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573  sngl bj-csngl 37009  tag bj-ctag 37018

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 2113  ax-9 2121  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2710  df-cleq 2723  df-clel 2806  df-rex 3057  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-pw 4549  df-sn 4574  df-pr 4576  df-bj-sngl 37010  df-bj-tag 37019

This theorem is referenced by: (None)