| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > unssad | Structured version Visualization version GIF version | ||
| Description: If (𝐴 ∪ 𝐵) is contained in 𝐶, so is 𝐴. One-way deduction form of unss 4190. Partial converse of unssd 4192. (Contributed by David Moews, 1-May-2017.) |
| Ref | Expression |
|---|---|
| unssad.1 | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) |
| Ref | Expression |
|---|---|
| unssad | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unssad.1 | . . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
| 2 | unss 4190 | . . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) ↔ (𝐴 ∪ 𝐵) ⊆ 𝐶) | |
| 3 | 1, 2 | sylibr 234 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶)) |
| 4 | 3 | simpld 494 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∪ cun 3949 ⊆ wss 3951 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 |
| This theorem is referenced by: naddcllem 8714 ersym 8757 findcard2d 9206 finsschain 9399 r0weon 10052 ackbij1lem16 10274 wunex2 10778 sumsplit 15804 fsumabs 15837 fsumiun 15857 mrieqvlemd 17672 yonedalem1 18317 yonedalem21 18318 yonedalem22 18323 yonffthlem 18327 lsmsp 21085 mplcoe1 22055 mdetunilem9 22626 ordtbas 23200 isufil2 23916 ufileu 23927 filufint 23928 fmfnfm 23966 flimclslem 23992 fclsfnflim 24035 flimfnfcls 24036 imasdsf1olem 24383 limcdif 25911 jensenlem1 27030 jensenlem2 27031 jensen 27032 gsumvsca1 33232 gsumvsca2 33233 qsdrngilem 33522 fldgenfldext 33718 evls1fldgencl 33720 fldextrspunlem1 33725 fldextrspunfld 33726 algextdeglem1 33758 algextdeglem2 33759 algextdeglem3 33760 algextdeglem4 33761 constrextdg2lem 33789 ordtconnlem1 33923 ssmcls 35572 mclsppslem 35588 rngunsnply 43181 mptrcllem 43626 clcnvlem 43636 brtrclfv2 43740 isotone1 44061 dvnprodlem1 45961 |
| Copyright terms: Public domain | W3C validator |