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Mirrors > Home > NFE Home > Th. List > ssrin | GIF version |
Description: Add right intersection to subclass relation. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
ssrin | ⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3267 | . . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B)) | |
2 | 1 | anim1d 547 | . . 3 ⊢ (A ⊆ B → ((x ∈ A ∧ x ∈ C) → (x ∈ B ∧ x ∈ C))) |
3 | elin 3219 | . . 3 ⊢ (x ∈ (A ∩ C) ↔ (x ∈ A ∧ x ∈ C)) | |
4 | elin 3219 | . . 3 ⊢ (x ∈ (B ∩ C) ↔ (x ∈ B ∧ x ∈ C)) | |
5 | 2, 3, 4 | 3imtr4g 261 | . 2 ⊢ (A ⊆ B → (x ∈ (A ∩ C) → x ∈ (B ∩ C))) |
6 | 5 | ssrdv 3278 | 1 ⊢ (A ⊆ B → (A ∩ C) ⊆ (B ∩ C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∈ wcel 1710 ∩ cin 3208 ⊆ wss 3257 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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 |
This theorem is referenced by: sslin 3481 ss2in 3482 ssdisj 3600 ssdifin0 3631 pw1ss 4169 ssres 4990 |
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