Theorem bdeq0 15077

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 15075	. . 3 ⊢ BOUNDED ∅
2	1	bdss 15074	. 2 ⊢ BOUNDED 𝑥 ⊆ ∅
3		0ss 3476	. . 3 ⊢ ∅ ⊆ 𝑥
4		eqss 3185	. . 3 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
5	3, 4	mpbiran2 943	. 2 ⊢ (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
6	2, 5	bd0r 15035	1 ⊢ BOUNDED 𝑥 = ∅

Syntax hints:   = wceq 1364   ⊆ wss 3144  ∅c0 3437  BOUNDED wbd 15022

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15023  ax-bdim 15024  ax-bdn 15027  ax-bdal 15028  ax-bdeq 15030

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-bdc 15051

This theorem is referenced by:  bj-bd0el  15078  bj-nn0suc0  15160