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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjss | Structured version Visualization version GIF version |
Description: Subclass theorem for disjoints. (Contributed by Peter Mazsa, 28-Oct-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
Ref | Expression |
---|---|
disjss | ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 5869 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
2 | funALTVss 38160 | . . . 4 ⊢ (◡𝐴 ⊆ ◡𝐵 → ( FunALTV ◡𝐵 → FunALTV ◡𝐴)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV ◡𝐵 → FunALTV ◡𝐴)) |
4 | relss 5777 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
5 | 3, 4 | anim12d 608 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (( FunALTV ◡𝐵 ∧ Rel 𝐵) → ( FunALTV ◡𝐴 ∧ Rel 𝐴))) |
6 | dfdisjALTV 38174 | . 2 ⊢ ( Disj 𝐵 ↔ ( FunALTV ◡𝐵 ∧ Rel 𝐵)) | |
7 | dfdisjALTV 38174 | . 2 ⊢ ( Disj 𝐴 ↔ ( FunALTV ◡𝐴 ∧ Rel 𝐴)) | |
8 | 5, 6, 7 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3944 ◡ccnv 5671 Rel wrel 5677 FunALTV wfunALTV 37668 Disj wdisjALTV 37671 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
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 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-coss 37872 df-cnvrefrel 37988 df-funALTV 38143 df-disjALTV 38166 |
This theorem is referenced by: disjssi 38193 disjssd 38194 |
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