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

Theorem bdssexg 10024
 Description: Bounded version of ssexg 3896. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdssexg.bd BOUNDED 𝐴
Assertion
Ref Expression
bdssexg ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Proof of Theorem bdssexg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sseq2 2967 . . . 4 (𝑥 = 𝐵 → (𝐴𝑥𝐴𝐵))
21imbi1d 220 . . 3 (𝑥 = 𝐵 → ((𝐴𝑥𝐴 ∈ V) ↔ (𝐴𝐵𝐴 ∈ V)))
3 bdssexg.bd . . . 4 BOUNDED 𝐴
4 vex 2560 . . . 4 𝑥 ∈ V
53, 4bdssex 10022 . . 3 (𝐴𝑥𝐴 ∈ V)
62, 5vtoclg 2613 . 2 (𝐵𝐶 → (𝐴𝐵𝐴 ∈ V))
76impcom 116 1 ((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  BOUNDED wbdc 9960 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-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-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961 This theorem is referenced by:  bdssexd  10025  bdrabexg  10026  bdunexb  10040
 Copyright terms: Public domain W3C validator