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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 3211 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | eqcoms 2180 | 1 ⊢ (𝐵 = 𝐴 → 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: disjeq2 3986 disjeq1 3989 poeq2 4302 seeq1 4341 seeq2 4342 dmcoeq 4901 xp11m 5069 funeq 5238 fconst3m 5738 tposeq 6251 undifdcss 6925 ennnfonelemk 12404 ennnfonelemss 12414 qnnen 12435 imasaddfnlemg 12741 topgele 13690 topontopn 13698 txdis 13938 |
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