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Mirrors > Home > ILE Home > Th. List > eqimss2i | GIF version |
Description: Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.) |
Ref | Expression |
---|---|
eqimssi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
eqimss2i | ⊢ 𝐵 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3018 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | eqimssi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
3 | 1, 2 | sseqtr4i 3032 | 1 ⊢ 𝐵 ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ⊆ wss 2973 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-in 2979 df-ss 2986 |
This theorem is referenced by: (None) |
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