New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > ssbrd | GIF version |
Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
ssbrd.1 | ⊢ (φ → A ⊆ B) |
Ref | Expression |
---|---|
ssbrd | ⊢ (φ → (CAD → CBD)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbrd.1 | . . 3 ⊢ (φ → A ⊆ B) | |
2 | 1 | sseld 3272 | . 2 ⊢ (φ → (〈C, D〉 ∈ A → 〈C, D〉 ∈ B)) |
3 | df-br 4640 | . 2 ⊢ (CAD ↔ 〈C, D〉 ∈ A) | |
4 | df-br 4640 | . 2 ⊢ (CBD ↔ 〈C, D〉 ∈ B) | |
5 | 2, 3, 4 | 3imtr4g 261 | 1 ⊢ (φ → (CAD → CBD)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ⊆ wss 3257 〈cop 4561 class class class wbr 4639 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-br 4640 |
This theorem is referenced by: ssbri 4681 coss1 4872 coss2 4873 |
Copyright terms: Public domain | W3C validator |