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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 3182 | . 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 1363 ⊆ wss 3141 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-11 1516 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-in 3147 df-ss 3154 |
This theorem is referenced by: eqimss2 3222 uneqin 3398 sssnr 3765 sssnm 3766 ssprr 3768 sstpr 3769 snsspw 3776 pwpwssunieq 3987 elpwuni 3988 disjeq2 3996 disjeq1 3999 pwne 4172 pwssunim 4296 poeq2 4312 seeq1 4351 seeq2 4352 trsucss 4435 onsucelsucr 4519 xp11m 5079 funeq 5248 fnresdm 5337 fssxp 5395 ffdm 5398 fcoi1 5408 fof 5450 dff1o2 5478 fvmptss2 5604 fvmptssdm 5613 fprg 5712 dff1o6 5790 tposeq 6262 el2oss1o 6458 nntri1 6511 nntri2or2 6513 nnsseleq 6516 infnninf 7136 infnninfOLD 7137 nninfwlpoimlemg 7187 exmidontri2or 7256 frec2uzf1od 10420 hashinfuni 10771 setsresg 12514 setsslid 12527 strle1g 12580 cncnpi 14024 hmeores 14111 limcimolemlt 14429 recnprss 14452 0nninf 15050 nninfall 15055 |
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