Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjss Structured version   Visualization version   GIF version

Theorem disjss 38689
Description: Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)
Assertion
Ref Expression
disjss (𝐴𝐵 → ( Disj 𝐵 → Disj 𝐴))

Proof of Theorem disjss
StepHypRef Expression
1 cnvss 5897 . . . 4 (𝐴𝐵𝐴𝐵)
2 funALTVss 38657 . . . 4 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
31, 2syl 17 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
4 relss 5805 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 608 . 2 (𝐴𝐵 → (( FunALTV 𝐵 ∧ Rel 𝐵) → ( FunALTV 𝐴 ∧ Rel 𝐴)))
6 dfdisjALTV 38671 . 2 ( Disj 𝐵 ↔ ( FunALTV 𝐵 ∧ Rel 𝐵))
7 dfdisjALTV 38671 . 2 ( Disj 𝐴 ↔ ( FunALTV 𝐴 ∧ Rel 𝐴))
85, 6, 73imtr4g 296 1 (𝐴𝐵 → ( Disj 𝐵 → Disj 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3976  ccnv 5699  Rel wrel 5705   FunALTV wfunALTV 38168   Disj wdisjALTV 38171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-coss 38369  df-cnvrefrel 38485  df-funALTV 38640  df-disjALTV 38663
This theorem is referenced by:  disjssi  38690  disjssd  38691
  Copyright terms: Public domain W3C validator