Theorem bdeq0 15597

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 15595	. . 3 ⊢ BOUNDED ∅
2	1	bdss 15594	. 2 ⊢ BOUNDED 𝑥 ⊆ ∅
3		0ss 3490	. . 3 ⊢ ∅ ⊆ 𝑥
4		eqss 3199	. . 3 ⊢ (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
5	3, 4	mpbiran2 943	. 2 ⊢ (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
6	2, 5	bd0r 15555	1 ⊢ BOUNDED 𝑥 = ∅

Syntax hints:   = wceq 1364   ⊆ wss 3157  ∅c0 3451  BOUNDED wbd 15542

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15543  ax-bdim 15544  ax-bdn 15547  ax-bdal 15548  ax-bdeq 15550

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3452  df-bdc 15571

This theorem is referenced by:  bj-bd0el  15598  bj-nn0suc0  15680