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Mirrors > Home > ILE Home > Th. List > eqimss2 | GIF version |
Description: Equality implies the subclass relation. (Contributed by NM, 23-Nov-2003.) |
Ref | Expression |
---|---|
eqimss2 | ⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3101 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | eqcoms 2103 | 1 ⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ⊆ wss 3021 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-11 1452 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-in 3027 df-ss 3034 |
This theorem is referenced by: disjeq2 3856 disjeq1 3859 poeq2 4160 seeq1 4199 seeq2 4200 dmcoeq 4747 xp11m 4913 funeq 5079 fconst3m 5571 tposeq 6074 undifdcss 6740 ennnfonelemk 11705 ennnfonelemss 11715 qnnen 11736 topgele 11978 topontopn 11986 txdis 12227 |
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