Theorem bdeq0 16876

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 16874	. . 3 ⊢ BOUNDED ∅
2	1	bdss 16873	. 2 ⊢ BOUNDED 𝑥 ⊆ ∅
3		0ss 3561	. . 3 ⊢ ∅ ⊆ 𝑥
4		eqss 3263	. . 3 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
5	3, 4	mpbiran2 954	. 2 ⊢ (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
6	2, 5	bd0r 16834	1 ⊢ BOUNDED 𝑥 = ∅

Syntax hints:   = wceq 1402   ⊆ wss 3220  ∅c0 3520  BOUNDED wbd 16821

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16822  ax-bdim 16823  ax-bdn 16826  ax-bdal 16827  ax-bdeq 16829

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-bdc 16850

This theorem is referenced by:  bj-bd0el  16877  bj-nn0suc0  16959