Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ssini | Structured version Visualization version GIF version |
Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.) |
Ref | Expression |
---|---|
ssini.1 | ⊢ 𝐴 ⊆ 𝐵 |
ssini.2 | ⊢ 𝐴 ⊆ 𝐶 |
Ref | Expression |
---|---|
ssini | ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssini.1 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
2 | ssini.2 | . . 3 ⊢ 𝐴 ⊆ 𝐶 | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ (𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) |
4 | ssin 3835 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | |
5 | 3, 4 | mpbi 220 | 1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∩ cin 3573 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 |
This theorem is referenced by: inv1 3970 hartogslem1 8447 xptrrel 13719 fbasrn 21688 limciun 23658 hlimcaui 28093 chdmm1i 28336 chm0i 28349 ledii 28395 lejdii 28397 mayetes3i 28588 mdslj2i 29179 mdslmd2i 29189 sumdmdlem2 29278 sigapildsys 30225 ssoninhaus 32447 bj-disj2r 33013 idinxpres 34088 icomnfinre 39779 fouriersw 40448 sge0split 40626 |
Copyright terms: Public domain | W3C validator |