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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 3185 | . 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 1364 ⊆ wss 3144 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-in 3150 df-ss 3157 |
This theorem is referenced by: eqimss2 3225 uneqin 3401 sssnr 3768 sssnm 3769 ssprr 3771 sstpr 3772 snsspw 3779 pwpwssunieq 3990 elpwuni 3991 disjeq2 3999 disjeq1 4002 pwne 4175 pwssunim 4299 poeq2 4315 seeq1 4354 seeq2 4355 trsucss 4438 onsucelsucr 4522 xp11m 5082 funeq 5251 fnresdm 5340 fssxp 5398 ffdm 5401 fcoi1 5411 fof 5453 dff1o2 5481 fvmptss2 5607 fvmptssdm 5616 fprg 5715 dff1o6 5793 tposeq 6266 el2oss1o 6462 nntri1 6515 nntri2or2 6517 nnsseleq 6520 infnninf 7140 infnninfOLD 7141 nninfwlpoimlemg 7191 exmidontri2or 7260 frec2uzf1od 10424 hashinfuni 10775 setsresg 12518 setsslid 12531 strle1g 12584 cncnpi 14125 hmeores 14212 limcimolemlt 14530 recnprss 14553 0nninf 15151 nninfall 15156 |
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