Theorem bj-tagss 37306

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 37301	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 37296	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5294	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5243	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4729	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 230	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4132	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3969	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2114   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  sngl bj-csngl 37291  tag bj-ctag 37300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3063  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-pw 4544  df-sn 4569  df-pr 4571  df-bj-sngl 37292  df-bj-tag 37301

This theorem is referenced by: (None)