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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 4204 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) ↔ 𝐴 ⊆ (𝐵 ∩ 𝐶)) | |
5 | 3, 4 | mpbi 231 | 1 ⊢ 𝐴 ⊆ (𝐵 ∩ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∩ cin 3932 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 |
This theorem is referenced by: inv1 4345 cnvrescnv 6045 hartogslem1 8994 xptrrel 14328 fbasrn 22420 limciun 24419 hlimcaui 28940 chdmm1i 29181 chm0i 29194 ledii 29240 lejdii 29242 mayetes3i 29433 mdslj2i 30024 mdslmd2i 30034 sumdmdlem2 30123 sigapildsys 31320 ssoninhaus 33693 bj-disj2r 34237 bj-idres 34344 bj-rvecsscvec 34473 icomnfinre 41704 fouriersw 42393 sge0split 42568 |
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