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Mirrors > Home > ILE Home > Th. List > sstr | GIF version |
Description: Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.) |
Ref | Expression |
---|---|
sstr | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr2 3006 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶)) | |
2 | 1 | imp 122 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ⊆ 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: sstrd 3009 sylan9ss 3012 ssdifss 3102 uneqin 3215 ssindif0im 3303 undifss 3323 ssrnres 4783 relrelss 4864 fco 5076 fssres 5086 ssimaex 5255 tpostpos2 5903 smores 5930 iccsupr 8989 |
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