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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjssi | Structured version Visualization version GIF version |
Description: Subclass theorem for disjoints, inference version. (Contributed by Peter Mazsa, 28-Sep-2021.) |
Ref | Expression |
---|---|
disjssi.1 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
disjssi | ⊢ ( Disj 𝐵 → Disj 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjssi.1 | . 2 ⊢ 𝐴 ⊆ 𝐵 | |
2 | disjss 36998 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( Disj 𝐵 → Disj 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ( Disj 𝐵 → Disj 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3898 Disj wdisjALTV 36472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-br 5093 df-opab 5155 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-coss 36678 df-cnvrefrel 36794 df-funALTV 36949 df-disjALTV 36972 |
This theorem is referenced by: disjimres 37017 disjimin 37018 |
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