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Theorem bdeq0 9302
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 9300 . . 3 BOUNDED
21bdss 9299 . 2 BOUNDED x ⊆ ∅
3 0ss 3249 . . 3 ∅ ⊆ x
4 eqss 2954 . . 3 (x = ∅ ↔ (x ⊆ ∅ ∅ ⊆ x))
53, 4mpbiran2 847 . 2 (x = ∅ ↔ x ⊆ ∅)
62, 5bd0r 9260 1 BOUNDED x = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wss 2911  c0 3218  BOUNDED wbd 9247
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9248  ax-bdim 9249  ax-bdn 9252  ax-bdal 9253  ax-bdeq 9255
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-bdc 9276
This theorem is referenced by:  bj-bd0el  9303  bj-nn0suc0  9384
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