Theorem ssbri 4107

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 4105	. 2 ⊢ (⊤ → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))
4	3	mptru 1384	1 ⊢ (𝐶𝐴𝐷 → 𝐶𝐵𝐷)

Syntax hints:   → wi 4  ⊤wtru 1376   ⊆ wss 3177   class class class wbr 4062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-in 3183  df-ss 3190  df-br 4063

This theorem is referenced by:  brel  4748  swoer  6678  swoord1  6679  swoord2  6680  ecopover  6750  ecopoverg  6753  endom  6884