Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssd Structured version   Visualization version   GIF version

Theorem ssd 38704
Description: A sufficient condition for a subclass relationship. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypothesis
Ref Expression
ssd.1 ((𝜑𝑥𝐴) → 𝑥𝐵)
Assertion
Ref Expression
ssd (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Proof of Theorem ssd
StepHypRef Expression
1 nfv 1845 . 2 𝑥𝜑
2 ssd.1 . 2 ((𝜑𝑥𝐴) → 𝑥𝐵)
31, 2ssdf 38699 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1992  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-ral 2917  df-in 3567  df-ss 3574
This theorem is referenced by:  funimassd  38873  icomnfinre  39158  fnlimfvre  39278  allbutfifvre  39279  limsupresico  39304  rrxsnicc  39795  meaiuninclem  39972  meaiininclem  39975  borelmbl  40125  smflimlem1  40254  smflimlem2  40255  smfpimbor1lem1  40280  smfpimbor1lem2  40281  smfsuplem1  40292
  Copyright terms: Public domain W3C validator