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Theorem disjss 38112
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 5865 . . . 4 (𝐴𝐵𝐴𝐵)
2 funALTVss 38080 . . . 4 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
31, 2syl 17 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
4 relss 5774 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 608 . 2 (𝐴𝐵 → (( FunALTV 𝐵 ∧ Rel 𝐵) → ( FunALTV 𝐴 ∧ Rel 𝐴)))
6 dfdisjALTV 38094 . 2 ( Disj 𝐵 ↔ ( FunALTV 𝐵 ∧ Rel 𝐵))
7 dfdisjALTV 38094 . 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 3943  ccnv 5668  Rel wrel 5674   FunALTV wfunALTV 37585   Disj wdisjALTV 37588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-coss 37792  df-cnvrefrel 37908  df-funALTV 38063  df-disjALTV 38086
This theorem is referenced by:  disjssi  38113  disjssd  38114
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