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Theorem bdeq0 15077
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 15075 . . 3 BOUNDED
21bdss 15074 . 2 BOUNDED 𝑥 ⊆ ∅
3 0ss 3476 . . 3 ∅ ⊆ 𝑥
4 eqss 3185 . . 3 (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
53, 4mpbiran2 943 . 2 (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
62, 5bd0r 15035 1 BOUNDED 𝑥 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wss 3144  c0 3437  BOUNDED wbd 15022
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15023  ax-bdim 15024  ax-bdn 15027  ax-bdal 15028  ax-bdeq 15030
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-bdc 15051
This theorem is referenced by:  bj-bd0el  15078  bj-nn0suc0  15160
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