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Theorem disjss 38636
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 5896 . . . 4 (𝐴𝐵𝐴𝐵)
2 funALTVss 38604 . . . 4 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
31, 2syl 17 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
4 relss 5804 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 608 . 2 (𝐴𝐵 → (( FunALTV 𝐵 ∧ Rel 𝐵) → ( FunALTV 𝐴 ∧ Rel 𝐴)))
6 dfdisjALTV 38618 . 2 ( Disj 𝐵 ↔ ( FunALTV 𝐵 ∧ Rel 𝐵))
7 dfdisjALTV 38618 . 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 3970  ccnv 5698  Rel wrel 5704   FunALTV wfunALTV 38115   Disj wdisjALTV 38118
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2712  df-cleq 2726  df-clel 2813  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-coss 38316  df-cnvrefrel 38432  df-funALTV 38587  df-disjALTV 38610
This theorem is referenced by:  disjssi  38637  disjssd  38638
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