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Theorem bdeq0 9987
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 9985 . . 3 BOUNDED
21bdss 9984 . 2 BOUNDED 𝑥 ⊆ ∅
3 0ss 3255 . . 3 ∅ ⊆ 𝑥
4 eqss 2960 . . 3 (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
53, 4mpbiran2 848 . 2 (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
62, 5bd0r 9945 1 BOUNDED 𝑥 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1243  wss 2917  c0 3224  BOUNDED wbd 9932
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdal 9938  ax-bdeq 9940
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-bdc 9961
This theorem is referenced by:  bj-bd0el  9988  bj-nn0suc0  10075
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