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Theorem disjss 38203
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 5875 . . . 4 (𝐴𝐵𝐴𝐵)
2 funALTVss 38171 . . . 4 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
31, 2syl 17 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
4 relss 5783 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 608 . 2 (𝐴𝐵 → (( FunALTV 𝐵 ∧ Rel 𝐵) → ( FunALTV 𝐴 ∧ Rel 𝐴)))
6 dfdisjALTV 38185 . 2 ( Disj 𝐵 ↔ ( FunALTV 𝐵 ∧ Rel 𝐵))
7 dfdisjALTV 38185 . 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 3947  ccnv 5677  Rel wrel 5683   FunALTV wfunALTV 37679   Disj wdisjALTV 37682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-coss 37883  df-cnvrefrel 37999  df-funALTV 38154  df-disjALTV 38177
This theorem is referenced by:  disjssi  38204  disjssd  38205
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