**Description:** A relationship between subclass and intersection. Similar to Exercise 9
of TakeutiZaring p. 18. (Contributed by NM, 17-May-1994)

Ref | Expression | ||
---|---|---|---|

Assertion | sseqin2 | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |

Step | Hyp | Ref | Expression |
---|---|---|---|

1 | df-ss | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 ) | |

2 | incom | ⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 ) | |

3 | 2 | eqeq1i | ⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐴 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |

4 | 1 3 | bitri | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐴 ) = 𝐴 ) |