Theorem bdeq0 16488

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 16486	. . 3 ⊢ BOUNDED ∅
2	1	bdss 16485	. 2 ⊢ BOUNDED 𝑥 ⊆ ∅
3		0ss 3533	. . 3 ⊢ ∅ ⊆ 𝑥
4		eqss 3242	. . 3 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
5	3, 4	mpbiran2 949	. 2 ⊢ (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
6	2, 5	bd0r 16446	1 ⊢ BOUNDED 𝑥 = ∅

Syntax hints:   = wceq 1397   ⊆ wss 3200  ∅c0 3494  BOUNDED wbd 16433

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16434  ax-bdim 16435  ax-bdn 16438  ax-bdal 16439  ax-bdeq 16441

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-bdc 16462

This theorem is referenced by:  bj-bd0el  16489  bj-nn0suc0  16571