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Mirrors > Home > ILE Home > Th. List > sseqtri | GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 28-Jul-1995.) |
Ref | Expression |
---|---|
sseqtr.1 | ⊢ A ⊆ B |
sseqtr.2 | ⊢ B = 𝐶 |
Ref | Expression |
---|---|
sseqtri | ⊢ A ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqtr.1 | . 2 ⊢ A ⊆ B | |
2 | sseqtr.2 | . . 3 ⊢ B = 𝐶 | |
3 | 2 | sseq2i 2964 | . 2 ⊢ (A ⊆ B ↔ A ⊆ 𝐶) |
4 | 1, 3 | mpbi 133 | 1 ⊢ A ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: sseqtr4i 2972 eqimssi 2993 abssi 3009 ssun2 3101 inssddif 3172 difdifdirss 3301 pwundifss 4013 unixpss 4394 0ima 4628 |
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