Theorem bj-tagss 34285

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 34280	. 2 ⊢ tag 𝐴 = (sngl 𝐴 ∪ {∅})
2		bj-snglss 34275	. . 3 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴
3		0elpw 5247	. . . 4 ⊢ ∅ ∈ 𝒫 𝐴
4		0ex 5202	. . . . 5 ⊢ ∅ ∈ V
5	4	snss 4710	. . . 4 ⊢ (∅ ∈ 𝒫 𝐴 ↔ {∅} ⊆ 𝒫 𝐴)
6	3, 5	mpbi 232	. . 3 ⊢ {∅} ⊆ 𝒫 𝐴
7	2, 6	unssi 4159	. 2 ⊢ (sngl 𝐴 ∪ {∅}) ⊆ 𝒫 𝐴
8	1, 7	eqsstri 3999	1 ⊢ tag 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∈ wcel 2108   ∪ cun 3932   ⊆ wss 3934  ∅c0 4289  𝒫 cpw 4537  {csn 4559  sngl bj-csngl 34270  tag bj-ctag 34279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-pw 4539  df-sn 4560  df-pr 4562  df-bj-sngl 34271  df-bj-tag 34280

This theorem is referenced by: (None)