Theorem ssini 1660

Description: An inference showing that the a subclass of two classes is a subclass of their intersection.

Hypotheses

Ref	Expression
ssini.1	⊢ A ⊆ B
ssini.2	⊢ A ⊆ C

Assertion

Ref	Expression
ssini	⊢ A ⊆ (B ∩ C)

Proof of Theorem ssini

Step	Hyp	Ref	Expression
1		ssini.1	. . 3 ⊢ A ⊆ B
2		ssini.2	. . 3 ⊢ A ⊆ C
3	1, 2	pm3.2i 234	. 2 ⊢ (A ⊆ B ∧ A ⊆ C)
4		ssin 1659	. 2 ⊢ ((A ⊆ B ∧ A ⊆ C) ↔ A ⊆ (B ∩ C))
5	3, 4	mpbi 164	1 ⊢ A ⊆ (B ∩ C)

Syntax hints:   ∧ wa 196   ∩ cin 1486   ⊆ wss 1487

This theorem is referenced by:  inv 1723  chm1 5378  chdmm1 5398  chm0 5411  ledi 5447

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

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