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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjss | Structured version Visualization version GIF version |
Description: Subclass theorem for disjoint elementhood. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
eldisjss | ⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssres2 5894 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (◡ E ↾ 𝐴) ⊆ (◡ E ↾ 𝐵)) | |
2 | 1 | disjssd 36608 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ( Disj (◡ E ↾ 𝐵) → Disj (◡ E ↾ 𝐴))) |
3 | df-eldisj 36582 | . 2 ⊢ ( ElDisj 𝐵 ↔ Disj (◡ E ↾ 𝐵)) | |
4 | df-eldisj 36582 | . 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
5 | 2, 3, 4 | 3imtr4g 299 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( ElDisj 𝐵 → ElDisj 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3881 E cep 5474 ◡ccnv 5565 ↾ cres 5568 Disj wdisjALTV 36131 ElDisj weldisj 36133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-br 5069 df-opab 5131 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-coss 36301 df-cnvrefrel 36407 df-funALTV 36557 df-disjALTV 36580 df-eldisj 36582 |
This theorem is referenced by: eldisjssi 36614 eldisjssd 36615 |
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