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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 3151 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | eqcoms 2142 | 1 ⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ⊆ wss 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: disjeq2 3910 disjeq1 3913 poeq2 4222 seeq1 4261 seeq2 4262 dmcoeq 4811 xp11m 4977 funeq 5143 fconst3m 5639 tposeq 6144 undifdcss 6811 ennnfonelemk 11913 ennnfonelemss 11923 qnnen 11944 topgele 12196 topontopn 12204 txdis 12446 |
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