Theorem ssbri 3806

Description: Inference from a subclass relationship of binary relations. (Contributed by NM, 28-Mar-2007.) (Revised by Mario Carneiro, 8-Feb-2015.)

Hypothesis

Ref	Expression
ssbri.1	⊢ 𝐴 ⊆ 𝐵

Assertion

Ref	Expression
ssbri	⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Proof of Theorem ssbri

Step	Hyp	Ref	Expression
1		ssbri.1	. . . 4 ⊢ 𝐴 ⊆ 𝐵
2	1	a1i 9	. . 3 ⊢ (⊤ → 𝐴 ⊆ 𝐵)
3	2	ssbrd 3805	. 2 ⊢ (⊤ → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
4	3	trud 1252	1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4  ⊤wtru 1244   ⊆ wss 2917   class class class wbr 3764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-br 3765

This theorem is referenced by:  brel  4392  swoer  6134  swoord1  6135  swoord2  6136  ecopover  6204  ecopoverg  6207  endom  6243