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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 3121 | . 2 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | 1 | eqcoms 2120 | 1 ⊢ (𝐵 = 𝐴 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ⊆ wss 3041 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-in 3047 df-ss 3054 |
This theorem is referenced by: disjeq2 3880 disjeq1 3883 poeq2 4192 seeq1 4231 seeq2 4232 dmcoeq 4781 xp11m 4947 funeq 5113 fconst3m 5607 tposeq 6112 undifdcss 6779 ennnfonelemk 11840 ennnfonelemss 11850 qnnen 11871 topgele 12123 topontopn 12131 txdis 12373 |
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