Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdeq0 GIF version

Theorem bdeq0 10925
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 10923 . . 3 BOUNDED
21bdss 10922 . 2 BOUNDED 𝑥 ⊆ ∅
3 0ss 3298 . . 3 ∅ ⊆ 𝑥
4 eqss 3023 . . 3 (𝑥 = ∅ ↔ (𝑥 ⊆ ∅ ∧ ∅ ⊆ 𝑥))
53, 4mpbiran2 883 . 2 (𝑥 = ∅ ↔ 𝑥 ⊆ ∅)
62, 5bd0r 10883 1 BOUNDED 𝑥 = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wss 2982  c0 3267  BOUNDED wbd 10870
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bd0 10871  ax-bdim 10872  ax-bdn 10875  ax-bdal 10876  ax-bdeq 10878
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268  df-bdc 10899
This theorem is referenced by:  bj-bd0el  10926  bj-nn0suc0  11012
  Copyright terms: Public domain W3C validator