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Theorem bdeq0 7241
 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
StepHypRef Expression
1 bdcnul 7239 . . 3 BOUNDED
21bdss 7238 . 2 BOUNDED x ⊆ ∅
3 0ss 3232 . . 3 ∅ ⊆ x
4 eqss 2937 . . 3 (x = ∅ ↔ (x ⊆ ∅ ∅ ⊆ x))
53, 4mpbiran2 836 . 2 (x = ∅ ↔ x ⊆ ∅)
62, 5bd0r 7199 1 BOUNDED x = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1228   ⊆ wss 2894  ∅c0 3201  BOUNDED wbd 7186 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7187  ax-bdim 7188  ax-bdn 7191  ax-bdal 7192  ax-bdeq 7194 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-dif 2897  df-in 2901  df-ss 2908  df-nul 3202  df-bdc 7215 This theorem is referenced by:  bj-bd0el  7242  bj-nn0suc0  7319
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