Theorem bj-tagss 37564

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 37559	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 37554	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5330	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5275	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4755	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 233	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4152	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3991	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2150   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  {csn 4594  sngl bj-csngl 37549  tag bj-ctag 37558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rex 3097  df-v 3464  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-pw 4569  df-sn 4595  df-pr 4597  df-bj-sngl 37550  df-bj-tag 37559

This theorem is referenced by: (None)