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Mirrors > Home > ILE Home > Th. List > eqimss | GIF version |
Description: Equality implies the subclass relation. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 21-Jun-2011.) |
Ref | Expression |
---|---|
eqimss | ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqss 3172 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
2 | 1 | simplbi 274 | 1 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ⊆ wss 3131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-in 3137 df-ss 3144 |
This theorem is referenced by: eqimss2 3212 uneqin 3388 sssnr 3755 sssnm 3756 ssprr 3758 sstpr 3759 snsspw 3766 pwpwssunieq 3977 elpwuni 3978 disjeq2 3986 disjeq1 3989 pwne 4162 pwssunim 4286 poeq2 4302 seeq1 4341 seeq2 4342 trsucss 4425 onsucelsucr 4509 xp11m 5069 funeq 5238 fnresdm 5327 fssxp 5385 ffdm 5388 fcoi1 5398 fof 5440 dff1o2 5468 fvmptss2 5594 fvmptssdm 5603 fprg 5702 dff1o6 5780 tposeq 6251 el2oss1o 6447 nntri1 6500 nntri2or2 6502 nnsseleq 6505 infnninf 7125 infnninfOLD 7126 nninfwlpoimlemg 7176 exmidontri2or 7245 frec2uzf1od 10409 hashinfuni 10760 setsresg 12503 setsslid 12516 strle1g 12568 cncnpi 13889 hmeores 13976 limcimolemlt 14294 recnprss 14317 0nninf 14915 nninfall 14920 |
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