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Theorem disjss 36465
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 5715 . . . 4 (𝐴𝐵𝐴𝐵)
2 funALTVss 36433 . . . 4 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
31, 2syl 17 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
4 relss 5627 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
53, 4anim12d 612 . 2 (𝐴𝐵 → (( FunALTV 𝐵 ∧ Rel 𝐵) → ( FunALTV 𝐴 ∧ Rel 𝐴)))
6 dfdisjALTV 36447 . 2 ( Disj 𝐵 ↔ ( FunALTV 𝐵 ∧ Rel 𝐵))
7 dfdisjALTV 36447 . 2 ( Disj 𝐴 ↔ ( FunALTV 𝐴 ∧ Rel 𝐴))
85, 6, 73imtr4g 299 1 (𝐴𝐵 → ( Disj 𝐵 → Disj 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wss 3843  ccnv 5524  Rel wrel 5530   FunALTV wfunALTV 35987   Disj wdisjALTV 35990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-coss 36160  df-cnvrefrel 36266  df-funALTV 36416  df-disjALTV 36439
This theorem is referenced by:  disjssi  36466  disjssd  36467
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