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Theorem bdeq0 11758
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 11756 . . 3 BOUNDED
21bdss 11755 . 2 BOUNDED 𝑥 ⊆ ∅
3 0ss 3321 . . 3 ∅ ⊆ 𝑥
4 eqss 3040 . . 3 (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
53, 4mpbiran2 887 . 2 (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
62, 5bd0r 11716 1 BOUNDED 𝑥 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wss 2999  c0 3286  BOUNDED wbd 11703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11704  ax-bdim 11705  ax-bdn 11708  ax-bdal 11709  ax-bdeq 11711
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287  df-bdc 11732
This theorem is referenced by:  bj-bd0el  11759  bj-nn0suc0  11845
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