Theorem disjss 39480

Description: Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.)

Assertion

Ref	Expression
disjss	⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴))

Proof of Theorem disjss

Step	Hyp	Ref	Expression
1		cnvss 5858	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)
2		funALTVss 39433	. . . 4 ⊢ (◡𝐴 ⊆ ◡𝐵 → ( FunALTV ◡𝐵 → FunALTV ◡𝐴))
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV ◡𝐵 → FunALTV ◡𝐴))
4		relss 5768	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))
5	3, 4	anim12d 620	. 2 ⊢ (𝐴 ⊆ 𝐵 → (( FunALTV ◡𝐵 ∧ Rel 𝐵) → ( FunALTV ◡𝐴 ∧ Rel 𝐴)))
6		dfdisjALTV 39447	. 2 ⊢ ( Disj 𝐵 ↔ ( FunALTV ◡𝐵 ∧ Rel 𝐵))
7		dfdisjALTV 39447	. 2 ⊢ ( Disj 𝐴 ↔ ( FunALTV ◡𝐴 ∧ Rel 𝐴))
8	5, 6, 7	3imtr4g 299	1 ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ⊆ wss 3905  ^◡ccnv 5660  Rel wrel 5666   FunALTV wfunALTV 38865   Disj wdisjALTV 38868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-coss 39150  df-cnvrefrel 39256  df-funALTV 39416  df-disjALTV 39439

This theorem is referenced by:  disjssi  39481  disjssd  39482