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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssbr | Structured version Visualization version GIF version |
Description: Subclass theorem for binary relation, in a more searchable form: (𝑅 ⊆ 𝑆 → (𝐴𝑅𝐵 → 𝐴𝑆𝐵)). (Contributed by Peter Mazsa, 11-Nov-2019.) |
Ref | Expression |
---|---|
ssbr | ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3597 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (〈𝐶, 𝐷〉 ∈ 𝐴 → 〈𝐶, 𝐷〉 ∈ 𝐵)) | |
2 | df-br 4654 | . 2 ⊢ (𝐶𝐴𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐴) | |
3 | df-br 4654 | . 2 ⊢ (𝐶𝐵𝐷 ↔ 〈𝐶, 𝐷〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3imtr4g 285 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ⊆ wss 3574 〈cop 4183 class class class wbr 4653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 df-br 4654 |
This theorem is referenced by: (None) |
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