Theorem bdeq0 16160

Description: Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)

Assertion

Ref	Expression
bdeq0	⊢ BOUNDED 𝑥 = ∅

Proof of Theorem bdeq0

Step	Hyp	Ref	Expression
1		bdcnul 16158	. . 3 ⊢ BOUNDED ∅
2	1	bdss 16157	. 2 ⊢ BOUNDED 𝑥 ⊆ ∅
3		0ss 3530	. . 3 ⊢ ∅ ⊆ 𝑥
4		eqss 3239	. . 3 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
5	3, 4	mpbiran2 947	. 2 ⊢ (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
6	2, 5	bd0r 16118	1 ⊢ BOUNDED 𝑥 = ∅

Syntax hints:   = wceq 1395   ⊆ wss 3197  ∅c0 3491  BOUNDED wbd 16105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16106  ax-bdim 16107  ax-bdn 16110  ax-bdal 16111  ax-bdeq 16113

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-bdc 16134

This theorem is referenced by:  bj-bd0el  16161  bj-nn0suc0  16243