Theorem bj-tagss 38070

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 38065	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 38060	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5321	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5264	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4745	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 233	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4192	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3959	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

This proof depends on syntax axioms:   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560  {csn 4588  sngl bj-csngl 38055  tag bj-ctag 38064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rex 3091  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591  df-bj-sngl 38056  df-bj-tag 38065

This theorem is used by: (None)